Hubbry Logo
search
logo
Quadratic equation avatar
Quadratic equation
Quadratic equation
current hub
Q

Quadratic equation

logo
Community Hub0 Subscribers
Read side by side
Quadratic equation
View on Wikipedia
from Wikipedia

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as[1] where the variable ⁠⁠ represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.[2]

The values of ⁠⁠ that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation[3] where r and s are the solutions for ⁠⁠.

The quadratic formula expresses the solutions in terms of a, b, and c. Completing the square is one of several ways for deriving the formula.

Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.[4][5]

The quadratic equation contains only powers of ⁠⁠ that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.

Solving the quadratic equation

[edit]
Figure 1. Plots of the quadratic function, y = eh x squared plus b x plus c, varying each coefficient separately while the other coefficients are fixed at values eh = 1, b = 0, c = 0. The left plot illustrates varying c. When c equals 0, the vertex of the parabola representing the quadratic function is centered on the origin, and the parabola rises on both sides of the origin, opening to the top. When c is greater than zero, the parabola does not change in shape, but its vertex is raised above the origin. When c is less than zero, the vertex of the parabola is lowered below the origin. The center plot illustrates varying b. When b is less than zero, the parabola representing the quadratic function is unchanged in shape, but its vertex is shifted to the right of and below the origin. When b is greater than zero, its vertex is shifted to the left of and below the origin. The vertices of the family of curves created by varying b follow along a parabolic curve. The right plot illustrates varying eh. When eh is positive, the quadratic function is a parabola opening to the top. When eh is zero, the quadratic function is a horizontal straight line. When eh is negative, the quadratic function is a parabola opening to the bottom.
Figure 1. Plots of quadratic function y = ax2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0)

A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots. When there is only one distinct root, it can be interpreted as two roots with the same value, called a double root. When there are no real roots, the coefficients can be considered as complex numbers with zero imaginary part, and the quadratic equation still has two complex-valued roots, complex conjugates of each-other with a non-zero imaginary part. A quadratic equation whose coefficients are arbitrary complex numbers always has two complex-valued roots which may or may not be distinct.

The solutions of a quadratic equation can be found by several alternative methods.

Factoring by inspection

[edit]

It may be possible to express a quadratic equation ax2 + bx + c = 0 as a product (px + q)(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic.

For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.[6]: 202–207  If one is given a quadratic equation in the form x2 + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule"[7] and is related to Vieta's formulas). As an example, x2 + 5x + 6 factors as (x + 3)(x + 2). The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.

Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.[6]: 207 

Completing the square

[edit]
Main article: Completing the square
Figure 2 illustrates an x y plot of the quadratic function f of x equals x squared minus x minus 2. The x-coordinate of the points where the graph intersects the x-axis, x equals −1 and x equals 2, are the solutions of the quadratic equation x squared minus x minus 2 equals zero.
Figure 2. For the quadratic function y = x2 − x − 2, the points where the graph crosses the x-axis, x = −1 and x = 2, are the solutions of the quadratic equation x2 − x − 2 = 0.

The process of completing the square makes use of the algebraic identity which represents a well-defined algorithm that can be used to solve any quadratic equation.[6]: 207  Starting with a quadratic equation in standard form, ax2 + bx + c = 0

  1. Divide each side by a, the coefficient of the squared term.
  2. Subtract the constant term c/a from both sides.
  3. Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square.
  4. Write the left side as a square and simplify the right side if necessary.
  5. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
  6. Solve each of the two linear equations.

We illustrate use of this algorithm by solving 2x2 + 4x − 4 = 0

The plus–minus symbol "±" indicates that both and are solutions of the quadratic equation.[8]

Quadratic formula and its derivation

[edit]
Main article: Quadratic formula

Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula.[9] The mathematical proof will now be briefly summarized.[10] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation: Taking the square root of both sides, and isolating x, gives:

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax2 + 2bx + c = 0 or ax2 − 2bx + c = 0 ,[11] where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.

A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.

A lesser known quadratic formula, as used in Muller's method, provides the same roots via the equation This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is c/a. It also follows from dividing the quadratic equation by giving solving this for and then inverting.

One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0.

When neither a nor c is zero, the equality between the standard quadratic formula and Muller's method, can be verified by cross multiplication, and similarly for the other choice of signs.

Reduced quadratic equation

[edit]

It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a, which is always possible since a is non-zero. This produces the reduced quadratic equation:[12]

where p = b/a and q = c/a. This monic polynomial equation has the same solutions as the original.

The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is

Discriminant

[edit]
Figure 3. This figure plots three quadratic functions on a single Cartesian plane graph to illustrate the effects of discriminant values. When the discriminant, delta, is positive, the parabola intersects the x-axis at two points. When delta is zero, the vertex of the parabola touches the x-axis at a single point. When delta is negative, the parabola does not intersect the x-axis at all.
Figure 3. Discriminant signs

In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta:[13] A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

  • If the discriminant is positive, then there are two distinct roots both of which are real numbers. For quadratic equations with rational coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrationals.
  • If the discriminant is zero, then there is exactly one real root sometimes called a repeated or double root or two equal roots.
  • If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots[14] which are complex conjugates of each other. In these expressions i is the imaginary unit.

Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

Geometric interpretation

[edit]
Visualisation of the complex roots of y = ax2 + bx + c: the parabola is rotated 180° about its vertex (orange). Its x-intercepts are rotated 90° around their mid-point, and the Cartesian plane is interpreted as the complex plane (green).[15]

The function f(x) = ax2 + bx + c is a quadratic function.[16] The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. If a > 0, the parabola has a minimum point and opens upward. If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x-coordinate of the vertex will be located at , and the y-coordinate of the vertex may be found by substituting this x-value into the function. The y-intercept is located at the point (0, c).

The solutions of the quadratic equation ax2 + bx + c = 0 correspond to the roots of the function f(x) = ax2 + bx + c, since they are the values of x for which f(x) = 0. If a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. If the discriminant is positive, the graph touches the x-axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x-axis.

Quadratic factorization

[edit]

The term is a factor of the polynomial if and only if r is a root of the quadratic equation It follows from the quadratic formula that In the special case b2 = 4ac where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as

Graphical solution

[edit]
Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation 2x2 + 4x − 4 = 0. Although the display shows only five significant figures of accuracy, the retrieved value of xc is 0.732050807569, accurate to twelve significant figures.
A quadratic function without real root: y = (x − 5)2 + 9. The "3" is the imaginary part of the x-intercept. The real part is the x-coordinate of the vertex. Thus the roots are 5 ± 3i.

The solutions of the quadratic equation may be deduced from the graph of the quadratic function which is a parabola.

If the parabola intersects the x-axis in two points, there are two real roots, which are the x-coordinates of these two points (also called x-intercept).

If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the contact point between the graph and parabola.

If the parabola does not intersect the x-axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be.[17]

Let h and k be respectively the x-coordinate and the y-coordinate of the vertex of the parabola (that is the point with maximal or minimal y-coordinate. The quadratic function may be rewritten Let d be the distance between the point of y-coordinate 2k on the axis of the parabola, and a point on the parabola with the same y-coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is h, and their imaginary part are ±d. That is, the roots are or in the case of the example of the figure

Avoiding loss of significance

[edit]

Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). In this context, the quadratic formula is not completely stable.

This occurs when the roots have different order of magnitude, or, equivalently, when b2 and b2 − 4ac are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, r, can be computed as where R is the root that is bigger in magnitude. This is equivalent to using the formula

using the plus sign if and the minus sign if

A second form of cancellation can occur between the terms b2 and 4ac of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.[11][18]

Examples and applications

[edit]
The trajectory of the cliff jumper is parabolic because horizontal displacement is a linear function of time , while vertical displacement is a quadratic function of time . As a result, the path follows quadratic equation , where and are horizontal and vertical components of the original velocity, a is gravitational acceleration and h is original height. The a value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).

The golden ratio is found as the positive solution of the quadratic equation

The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.

Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation.

The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.

Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.

The equation given by Fuss' theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral.

Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.

In physics, for motion with constant acceleration , the displacement or position of a moving body can be expressed as a quadratic function of time given the initial position and initial velocity : .

In chemistry, the pH of a solution of weak acid can be calculated from the negative base-10 logarithm of the positive root of a quadratic equation in terms of the acidity constant and the analytical concentration of the acid.

History

[edit]

Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the Third Dynasty of Ur.[19] In modern notation, the problems typically involved solving a pair of simultaneous equations of the form: which is equivalent to the statement that x and y are the roots of the equation:[20]: 86 

The steps given by Babylonian scribes for solving the above rectangle problem, in terms of x and y, were as follows:

  1. Compute half of p.
  2. Square the result.
  3. Subtract q.
  4. Find the (positive) square root using a table of squares.
  5. Add together the results of steps (1) and (4) to give x.

In modern notation this means calculating , which is equivalent to the modern day quadratic formula for the larger real root (if any) with a = 1, b = −p, and c = q.

Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation.[21] Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.[22][23] Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics.[23][24] These early geometric methods do not appear to have had a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.[25]

In 628 AD, Brahmagupta, an Indian mathematician, gave in his book Brāhmasphuṭasiddhānta the first explicit (although still not completely general) solution of the quadratic equation ax2 + bx = c as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."[26] This is equivalent to The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as linear indeterminate equations (originally of type ax/c = y).

Muhammad ibn Musa al-Khwarizmi (9th century) developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process.[27] He also described the method of completing the square and recognized that the discriminant must be positive,[27][28]: 230  which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.[28]: 234  While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions,[27]: 191  as well as irrational numbers as solutions.[29] Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.[30] The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.[31]

The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.[32] His solution was largely based on Al-Khwarizmi's work.[27] The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.[33] By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594.[34] In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today.

Advanced topics

[edit]

Alternative methods of root calculation

[edit]

Vieta's formulas

[edit]
Main article: Vieta's formulas

Vieta's formulas (named after François Viète) are the relations between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation with the equation

The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving Also, these formulas for the vertex can be deduced directly from the formula (see Completing the square)

For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If |x2| << |x1|, then x1 + x2 ≈ x1, and we have the estimate: The second Vieta's formula then provides: These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. The figure shows the difference between[clarification needed] (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient b increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.

This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see Step response).

Trigonometric solution

[edit]

In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.[35] Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations.

It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation,

where the sign of the ± symbol is chosen so that a and c may both be positive. By substituting

and then multiplying through by cos2(θ) / c, we obtain

Introducing functions of 2θ and rearranging, we obtain

where the subscripts n and p correspond, respectively, to the use of a negative or positive sign in equation [1]. Substituting the two values of θn or θp found from equations [4] or [5] into [2] gives the required roots of [1]. Complex roots occur in the solution based on equation [5] if the absolute value of sin 2θp exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone.[36] Calculating complex roots would require using a different trigonometric form.[37]

To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy:

  1. A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries.
  7. (rounded to six significant figures)

Solution for complex roots in polar coordinates

[edit]

If the quadratic equation with real coefficients has two complex roots—the case where requiring a and c to have the same sign as each other—then the solutions for the roots can be expressed in polar form as[38]

where and

Geometric solution

[edit]
Figure 6. Geometric solution of eh x squared plus b x plus c = 0 using Lill's method. The geometric construction is as follows: Draw a trapezoid S Eh B C. Line S Eh of length eh is the vertical left side of the trapezoid. Line Eh B of length b is the horizontal bottom of the trapezoid. Line B C of length c is the vertical right side of the trapezoid. Line C S completes the trapezoid. From the midpoint of line C S, draw a circle passing through points C and S. Depending on the relative lengths of eh, b, and c, the circle may or may not intersect line Eh B. If it does, then the equation has a solution. If we call the intersection points X 1 and X 2, then the two solutions are given by negative Eh X 1 divided by S Eh, and negative Eh X 2 divided by S Eh.
Figure 6. Geometric solution of ax2 + bx + c = 0 using Lill's method. Solutions are −AX1/SA, −AX2/SA

The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.[39]

Carlyle circle of the quadratic equation x2 − sx + p = 0.

The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis.[40] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.

Generalization of quadratic equation

[edit]

The formula and its derivation remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2a is zero and it is impossible to divide by it.)

The symbol in the formula should be understood as "either of the two elements whose square is b2 − 4ac, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.

Characteristic 2

[edit]

In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root, so the solution is and there is only one root since In summary, See quadratic residue for more information about extracting square roots in finite fields.

In the case that b ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R(c) of c to be a root of the polynomial x2 + x + c, an element of the splitting field of that polynomial. One verifies that R(c) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax2 + bx + c are and

For example, let a denote a multiplicative generator of the group of units of F4, the Galois field of order four (thus a and a + 1 are roots of x2 + x + 1 over F4. Because (a + 1)2 = a, a + 1 is the unique solution of the quadratic equation x2 + a = 0. On the other hand, the polynomial x2 + ax + 1 is irreducible over F4, but it splits over F16, where it has the two roots ab and ab + a, where b is a root of x2 + x + a in F16.

This is a special case of Artin–Schreier theory.

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Quadratic equation

View on Grokipedia
from Grokipedia
A quadratic equation is a second-degree polynomial equation of the form $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are real numbers and $ a \neq 0 $.[1] This equation represents a fundamental concept in algebra, with solutions known as roots that can be real, repeated, or complex depending on the coefficients.[2] The graph of the corresponding quadratic function $ f(x) = ax^2 + bx + c $ is a parabola, which opens upward if $ a > 0 $ or downward if $ a < 0 $, and the roots correspond to the x-intercepts.[3] The history of quadratic equations dates back to ancient civilizations, with the Babylonians around 1800 BC developing algorithmic methods, such as completing the square, to solve problems that translate to quadratics, often in geometric contexts like finding lengths.[4] Euclid in approximately 300 BC employed geometrical techniques to determine roots equivalent to those of quadratic equations, though without modern algebraic notation.[4] Significant advancements occurred in India with Brahmagupta (598–665 AD), who provided a general solution incorporating negative quantities, and in the Islamic world with al-Khwarizmi around 820 AD, who classified six cases of quadratics and offered numerical and geometric proofs, excluding negatives and zero.[4] By the 12th century, Abraham bar Hiyya introduced complete solutions to Europe, and the modern quadratic formula was derived algebraically by Leonhard Euler in 1770.[5] Solutions to quadratic equations can be found through methods like factoring, completing the square, or the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $.[2] The expression $ b^2 - 4ac $, known as the discriminant, determines the nature of the roots: positive for two distinct real roots, zero for one real root (a repeated root), and negative for two complex conjugate roots.[6] These methods build on historical geometric approaches but leverage symbolic algebra for efficiency.[5] Quadratic equations have broad applications across fields, including physics for modeling projectile motion where distance fallen follows $ s = \frac{1}{2}gt^2 + v_0 t + s_0 $,[7] engineering for optimization problems like maximizing area or profit,[8] and computer graphics for calculating lines of sight to curved surfaces.[9] In economics, they help determine break-even points,[8] while in biology, they model population growth or enzyme kinetics under quadratic constraints.[10] Their versatility underscores their enduring importance in mathematics and science.

Definition and Basic Properties

Standard Form

A quadratic equation is an algebraic equation of the second degree with one unknown variable, expressed in its standard form as $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are real coefficients and $ a \neq 0 $.[11][12] This form represents a polynomial equation where the highest power of the variable $ x $ is 2, distinguishing it from linear (degree 1) or cubic (degree 3) equations.[11] The designation "quadratic" originates from the Latin term quadratus, the past participle of quadrare, meaning "to square," which alludes to the squared term $ x^2 $ central to the equation's structure.[13] The coefficient $ a $ scales the quadratic term and determines the parabola's orientation when graphed, while $ b $ and $ c $ adjust the linear and constant components, respectively.[14] Although the focus for solving quadratic equations remains the standard form, the related quadratic function can be expressed in vertex form as $ y = a(x - h)^2 + k $, where $ (h, k) $ identifies the parabola's vertex, aiding in graphical analysis.[15]

Coefficients and Discriminant

In the standard form of a quadratic equation, $ ax^2 + bx + c = 0 $, the coefficients $ a $, $ b $, and $ c $ (with $ a \neq 0 $) play distinct roles in defining the equation's graph as a parabola and its solution properties.[16] The coefficient $ a $ determines the direction and scaling of the parabola: if $ a > 0 $, the parabola opens upward; if $ a < 0 $, it opens downward, reflecting the graph across the x-axis. The magnitude of $ a $ affects the width, with larger $ |a| $ values narrowing the parabola and smaller values widening it.[16] The coefficient $ b $ influences the horizontal position of the parabola by setting the axis of symmetry at $ x = -\frac{b}{2a} $, which locates the vertex and turning point.[16] The constant term $ c $ represents the y-intercept, shifting the parabola vertically so that it crosses the y-axis at $ (0, c) $.[16] A key property derived from these coefficients is the discriminant, defined as $ D = b^2 - 4ac $, which appears under the square root in the quadratic formula and determines the nature of the roots without solving the equation.[17] The value of the discriminant classifies the roots as follows: if $ D > 0 $, there are two distinct real roots, corresponding to the parabola intersecting the x-axis at two points; if $ D = 0 $, there is exactly one real root (repeated), meaning the parabola touches the x-axis at its vertex; if $ D < 0 $, there are two complex conjugate roots, and the parabola does not intersect the x-axis.[17] Additionally, Vieta's formulas connect the coefficients to the roots: for roots $ r_1 $ and $ r_2 $, the sum $ r_1 + r_2 = -\frac{b}{a} $ and the product $ r_1 r_2 = \frac{c}{a} $, providing symmetric relations that highlight the interplay among $ a $, $ b $, and $ c $.[18]

Algebraic Solution Methods

Factoring by Inspection

Factoring by inspection is an algebraic technique for solving quadratic equations of the form $ ax^2 + bx + c = 0 $ by expressing the quadratic as a product of two linear factors $ (px + q)(rx + s) = 0 $, where $ pr = a $, $ qs = c $, and $ ps + qr = b $.[19] This method relies on identifying suitable integer or rational factors through trial and error or systematic search, leveraging the zero-factor property to find the roots as $ x = -q/p $ and $ x = -s/r $.[20] When the leading coefficient $ a = 1 $, the process simplifies to finding two numbers that multiply to $ c $ and add to $ b $. For example, in the equation $ x^2 + 5x + 6 = 0 $, the numbers 2 and 3 satisfy $ 2 \times 3 = 6 $ and $ 2 + 3 = 5 $, yielding the factorization $ (x + 2)(x + 3) = 0 $ with roots $ x = -2 $ and $ x = -3 $.[19] For cases where $ a \neq 1 $, the AC method is commonly used: first, identify two numbers that multiply to $ ac $ and add to $ b $, then rewrite the middle term and factor by grouping. Consider $ 2x^2 + 7x + 3 = 0 $; here, $ ac = 6 $, and the numbers 6 and 1 multiply to 6 and add to 7, so rewrite as $ 2x^2 + 6x + x + 3 = 0 $, group as $ (2x^2 + 6x) + (x + 3) = 0 $, factor to $ 2x(x + 3) + 1(x + 3) = 0 $, and obtain $ (2x + 1)(x + 3) = 0 $ with roots $ x = -1/2 $ and $ x = -3 $.[20] A quadratic equation can be factored into linear factors over the real numbers if its discriminant $ b^2 - 4ac $ is positive, indicating two distinct real roots, or zero, indicating a repeated real root.[21] This method works best with integer coefficients and rational roots, as the factors are typically integers in such cases.[19] The primary advantage of factoring by inspection is that it provides exact roots directly without invoking a general formula, making it intuitive for simple polynomials and useful in educational settings as an introductory solving technique.[19] However, it has limitations, particularly with non-integer coefficients, where finding suitable factors becomes trial-intensive or impractical, and it fails entirely for quadratics without rational roots even if real roots exist.[20]

Completing the Square

Completing the square is an algebraic technique for solving quadratic equations of the form $ ax^2 + bx + c = 0 $ by rewriting the expression as a difference of a perfect square trinomial and a constant, facilitating the extraction of roots via square roots. This method is particularly useful when the quadratic does not factor easily over the integers and provides insight into the equation's structure by transforming it into a form equivalent to the vertex representation of a parabola.[22] The origins of completing the square trace back to Old Babylonian mathematics around 1800 BCE, where it was employed geometrically to solve quadratic problems, such as completing L-shaped figures into squares on clay tablets like YBC 6967.[5] In the 9th century, the Persian scholar Muhammad ibn Musa al-Khwarizmi systematized the approach in his Compendium on Calculation by Completion and Balancing, presenting it as a core method for three cases of quadratics through geometric constructions, without considering negative roots.[23] Brahmagupta's 7th-century algebraic solutions to quadratics preceded al-Khwarizmi's geometric systematization of completing the square, which together influenced later European developments, such as those by Fibonacci in the 13th century.[5] Geometrically, completing the square can be demonstrated using areas of squares and rectangles. For the equation $ x^2 + bx = c $, construct a square of side length $ x $ to represent the $ x^2 $ term. Attach two rectangles, each with dimensions $ x \times \frac{b}{2} $, to two adjacent sides of the square, and place a square of side $ \frac{b}{2} $ in the corner to complete a larger square with side length $ x + \frac{b}{2} $. The area of this larger square is $ \left(x + \frac{b}{2}\right)^2 = x^2 + bx + \left(\frac{b}{2}\right)^2 = c + \left(\frac{b}{2}\right)^2 $, thus completing the square geometrically and illustrating the connection between algebraic manipulation and geometric areas.[24] To solve a quadratic equation using completing the square, follow these steps for the general form $ ax^2 + bx + c = 0 $, assuming $ a \neq 0 $:
  1. Divide both sides by $ a $ to obtain $ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $, making the leading coefficient 1.[22]
  2. Move the constant term to the right side: $ x^2 + \frac{b}{a}x = -\frac{c}{a} $.[22]
  3. Add $ \left( \frac{b}{2a} \right)^2 $ to both sides to complete the square on the left: $ x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2 $. The left side factors as $ \left( x + \frac{b}{2a} \right)^2 $.[22]
  4. Take the square root of both sides: $ x + \frac{b}{2a} = \pm \sqrt{ -\frac{c}{a} + \left( \frac{b}{2a} \right)^2 } $.[22]
  5. Solve for $ x $: $ x = -\frac{b}{2a} \pm \sqrt{ \left( \frac{b}{2a} \right)^2 - \frac{c}{a} } $.[22]
This process yields the roots provided the discriminant $ \left( \frac{b}{2a} \right)^2 - \frac{c}{a} \geq 0 $. Results can be verified by factoring if possible.[22] Completing the square and the quadratic formula both solve any quadratic equation $ ax^2 + bx + c = 0 $, yielding identical results. Completing the square is particularly useful when rewriting the quadratic in vertex form $ y = a(x - h)^2 + k $ is needed—for graphing the parabola, identifying the vertex, or solving optimization problems—or when deriving the quadratic formula. It is also advantageous when the leading coefficient $ a = 1 $ and the coefficients allow a straightforward process without introducing fractions (e.g., when $ b $ is even), providing deeper insight into the parabola's geometry beyond merely finding roots. In contrast, the quadratic formula offers a quick and reliable direct method, especially for equations with messy or non-integer coefficients.[25][26] The method also derives the vertex form of a quadratic function $ y = ax^2 + bx + c $, which is $ y = a(x - h)^2 + k $, where $ (h, k) $ is the vertex. Start by isolating the quadratic and linear terms:
y=ax2+bx+c y = a x^2 + b x + c
Factor out $ a $ from the first two terms:
y=a(x2+bax)+c y = a \left( x^2 + \frac{b}{a} x \right) + c
Complete the square inside the parentheses by adding and subtracting $ \left( \frac{b}{2a} \right)^2 $:
y=a(x2+bax+(b2a)2−(b2a)2)+c=a((x+b2a)2−(b2a)2)+c y = a \left( x^2 + \frac{b}{a} x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \right) + c = a \left( \left( x + \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \right) + c
Distribute $ a $ and simplify:
y=a(x+b2a)2−a(b2a)2+c=a(x−(−b2a))2+(c−b24a) y = a \left( x + \frac{b}{2a} \right)^2 - a \left( \frac{b}{2a} \right)^2 + c = a \left( x - \left( -\frac{b}{2a} \right) \right)^2 + \left( c - \frac{b^2}{4a} \right)
Thus, $ h = -\frac{b}{2a} $ and $ k = c - \frac{b^2}{4a} $, confirming the equivalence to the standard form.[27] For example, consider solving $ 2x^2 + 4x - 6 = 0 $. Divide by 2: $ x^2 + 2x - 3 = 0 $. Move the constant: $ x^2 + 2x = 3 $. Add $ (1)^2 = 1 $ to both sides: $ x^2 + 2x + 1 = 4 $, or $ (x + 1)^2 = 4 $. Take the square root: $ x + 1 = \pm 2 $, so $ x = -1 + 2 = 1 $ or $ x = -1 - 2 = -3 $. The roots are $ x = 1 $ and $ x = -3 $.[22] Completing the square serves as an intermediate step in deriving the quadratic formula, where the square root expression is further simplified to express the roots in terms of $ a $, $ b $, and $ c $ explicitly.[5]

Quadratic Formula and Derivation

The quadratic formula provides a universal algebraic method to find the roots of any quadratic equation of the form $ ax^2 + bx + c = 0 $, where $ a \neq 0 $, $ b $, and $ c $ are real coefficients. The solutions, or roots, are given by
x=−b±b2−4ac2a. x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
[28] This closed-form expression derives from the method of completing the square and applies regardless of whether the equation factors easily over the integers.[29] To derive the formula, begin with the general equation $ ax^2 + bx + c = 0 $. First, divide through by $ a $ to obtain the monic form:
x2+bax+ca=0. x^2 + \frac{b}{a}x + \frac{c}{a} = 0.
[28] Isolate the quadratic and linear terms:
x2+bax=−ca. x^2 + \frac{b}{a}x = -\frac{c}{a}.
To complete the square, add $ \left( \frac{b}{2a} \right)^2 $ to both sides:
x2+bax+(b2a)2=(b2a)2−ca. x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = \left( \frac{b}{2a} \right)^2 - \frac{c}{a}.
The left side factors as a perfect square:
(x+b2a)2=b2−4ac4a2. \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}.
Taking the square root of both sides yields:
x+b2a=±b2−4ac4a2=±b2−4ac2a. x + \frac{b}{2a} = \pm \sqrt{ \frac{b^2 - 4ac}{4a^2} } = \pm \frac{ \sqrt{b^2 - 4ac} }{2a}.
Solving for $ x $ gives the quadratic formula:
x=−b±b2−4ac2a. x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}.
[28] The quadratic formula provides a quick, reliable, and universal method for finding the roots of any quadratic equation $ ax^2 + bx + c = 0 $, especially when the coefficients are messy, non-integer, or do not lend themselves to easy factoring or completing the square. It serves as a direct plug-in method that always works. Since the quadratic formula is derived from completing the square, both methods yield identical results.[30] This formula is equivalent to solving by factoring, as the roots it provides directly correspond to the factors in the form $ a(x - r_1)(x - r_2) = 0 $, where $ r_1 $ and $ r_2 $ are the solutions from the formula.[31] For instance, if the roots are distinct real numbers, expanding the factored form recovers the original quadratic, confirming the methods yield identical solutions.[31] The nature of the real roots depends on the discriminant $ D = b^2 - 4ac $. If $ D > 0 $, there are two distinct real roots. If $ D = 0 $, there is exactly one real root (repeated). If $ D < 0 $, there are no real roots.[29] As an example, consider the equation $ 2x^2 + 5x - 3 = 0 $. Here, $ a = 2 $, $ b = 5 $, $ c = -3 $, so $ D = 25 + 24 = 49 > 0 $. The roots are
x=−5±494=−5±74, x = \frac{ -5 \pm \sqrt{49} }{4} = \frac{ -5 \pm 7 }{4},
yielding $ x = \frac{1}{2} $ and $ x = -3 $. These match the factored form $ (2x - 1)(x + 3) = 0 $.[28][31]

Geometric and Graphical Solutions

Parabola Interpretation

The graph of a quadratic function $ y = ax^2 + bx + c $, where $ a \neq 0 $, is a parabola.[32] If $ a > 0 $, the parabola opens upward, indicating a minimum value at the vertex; if $ a < 0 $, it opens downward, indicating a maximum value.[33] This U-shaped curve is symmetric and extends infinitely in the direction of its opening.[32] Geometrically, the parabola is defined as the locus of points equidistant from a fixed point (the focus) and a fixed straight line (the directrix). This focus-directrix definition is equivalent to the quadratic equation form in a coordinate system.[34] The vertex of the parabola represents its turning point and can be found using the formulas $ x = -\frac{b}{2a} $ for the x-coordinate and $ y = c - \frac{b^2}{4a} $ for the y-coordinate.[32] The axis of symmetry is the vertical line $ x = -\frac{b}{2a} $, which passes through the vertex and divides the parabola into two mirror-image halves.[33] The y-intercept occurs at the point $ (0, c) $, where the parabola crosses the y-axis.[35] The x-intercepts, or points where the parabola crosses the x-axis, correspond to the real roots of the quadratic equation $ ax^2 + bx + c = 0 $, obtained by setting $ y = 0 $.[32] These intercepts lie symmetrically about the axis of symmetry if two exist, providing visual insight into the number and location of solutions.[33] The standard parabola $ y = x^2 $ serves as the parent function, and the general form arises through transformations: a vertical stretch or compression by $ |a| $, a reflection over the x-axis if $ a < 0 $, a horizontal shift by $ -\frac{b}{2a} $ units, and a vertical shift by $ c $ units.[32] These transformations preserve the parabolic shape while altering its position, orientation, and scale.[33]

Geometric Constructions

There is a close connection between geometry and quadratic equations, as many geometric problems naturally lead to quadratic equations, and historically, such equations were solved geometrically using ruler and compass constructions in ancient Greek mathematics. The roots of quadratic equations can be constructed geometrically using a ruler (straightedge) and compass, methods that trace their origins to ancient Greek mathematics and form the basis of Euclidean geometry. These constructions transform the algebraic problem into finding specific lengths on a plane through intersections of lines and circles, where the coefficients are represented as given lengths. In Euclid's Elements, particularly Book II, such techniques are developed through propositions that geometrically interpret completing the square and extracting square roots, allowing solutions to specific quadratic forms without symbolic algebra. For instance, Proposition II.6 provides a construction for equations of the form x2−ax=b2x^2 - ax = b^2 by erecting a perpendicular of length bb at the midpoint of a segment of length aa and drawing a circle of radius a/2a/2 centered at one endpoint, with the intersection point yielding a length related to the root via the difference of squares: (a2)2−(a2−x)2=b2\left(\frac{a}{2}\right)^2 - \left(\frac{a}{2} - x\right)^2 = b^2.[36] Completing the square also has a direct geometric visualization using areas of squares and rectangles for equations like x2+bx=cx^2 + bx = c. By constructing a large square of side x+b/2x + b/2 composed of a central square of side xx, two rectangles each of area x⋅(b/2)x \cdot (b/2), and a small square of side b/2b/2, the area equality demonstrates the completion: (x+b/2)2=x2+bx+(b/2)2=c+(b/2)2(x + b/2)^2 = x^2 + bx + (b/2)^2 = c + (b/2)^2. Quadratic equations frequently arise from geometric intersections, such as between a straight line and a circle. Substituting the equation of the line into the circle's equation produces a quadratic whose solutions correspond to the coordinates of the intersection points. A notable historical example is the construction of the golden ratio, which satisfies the quadratic equation x2−x−1=0x^2 - x - 1 = 0 (or x2=x+1x^2 = x + 1), achieved geometrically in Euclid's Elements, Book VI, Proposition 30.[37] To address the general equation ax2+bx+c=0ax^2 + bx + c = 0, first reduce it to the monic depressed form x2+px+q=0x^2 + px + q = 0 by constructing the ratios p=b/ap = b/a and q=c/aq = c/a. This division is performed using similar triangles: draw a line segment of length aa, erect a perpendicular of length bb at one end, and draw a parallel line from a unit length to intersect, yielding pp as the intercepted segment. The same applies for qq. With pp and qq as lengths, construct perpendiculars to a base line to position coefficients spatially, then draw circles whose radii or diameters incorporate these lengths. Intersections with the base line or other constructed lines provide the root lengths, relying on Euclidean propositions for bisecting segments (Book I, Prop. 10), erecting perpendiculars (Book I, Prop. 11), and circle properties (Book III, Prop. 31).[38] A specific example is the geometric solution for x2+px+q=0x^2 + px + q = 0 using a circle with a diameter determined by the coefficients, known as the Carlyle circle. Draw a base line LL and select origin OO on LL. Erect a perpendicular to LL at OO, marking point AA at unit distance 1 from OO. From OO, along LL in the negative direction, mark point BB at distance ∣p∣|p| from OO. From BB, erect a perpendicular to LL in the same direction as OAOA if q>0q > 0 (or opposite if q<0q < 0), marking point CC at distance ∣q∣|q| from BB. Construct the circle with diameter ACAC: first, find the midpoint of ACAC as center using bisection, then set the radius to half ACAC with the compass. This circle intersects LL at two points R1R_1 and R2R_2 (potentially including OO if a root is zero), where the directed distances from OO to R1R_1 and R2R_2 are the roots of the equation. The method works because the circle's defining equation, when restricted to the base line, simplifies to the quadratic via the diameter endpoint property and the right-angle theorem in a semicircle (Euclid Book III, Prop. 31). For real roots, the discriminant must be non-negative, ensuring two intersections on LL; complex roots do not yield real intersections.[39] These constructions are limited to producing constructible numbers, which are real numbers obtainable from the rationals via a finite tower of quadratic field extensions using the given coefficient lengths as starting points. Roots of quadratics with constructible coefficients are always constructible, as solving x2+px+q=0x^2 + px + q = 0 involves at most one square root extraction, corresponding to a single quadratic extension. However, this restricts applicability to problems where solutions lie in such fields; for instance, roots requiring cubic or higher odd-degree extensions (as in angle trisection or cube duplication) cannot be constructed solely with ruler and compass, a result formalized in Galois theory.[40]

Graphical Methods

Graphical methods for solving quadratic equations involve plotting the quadratic function and identifying the points where the graph intersects the x-axis, which correspond to the real roots of the equation. To apply this approach, one graphs the equation $ y = ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants, and the x-intercepts provide the approximate values of the roots by visual inspection or measurement on the graph.[41][42] This method leverages the parabolic shape of the graph, allowing users to visualize the behavior of the quadratic.[43] The vertex of the parabola and its axis of symmetry play key roles in estimating roots more effectively. The vertex represents the turning point, and the axis of symmetry is the vertical line through the vertex, which bisects the parabola; roots, if real, lie symmetric about this axis, enabling quicker approximations of their positions relative to the vertex.[43] For instance, if the vertex is known to be at a certain x-value, one can estimate the roots by considering equal distances on either side where the graph crosses the x-axis. Digital tools enhance the precision of graphical solutions. Interactive graphing calculators like Desmos allow users to plot the quadratic function and zoom in on x-intercepts or use built-in features to detect exact intersection points, bridging the gap between visual approximation and algebraic accuracy.[44] These tools facilitate exploration of how changes in coefficients affect root locations. This method offers advantages in building intuition, as it visually reveals the number of real roots (zero, one, or two) and their approximate locations without complex calculations, making it accessible for initial explorations.[45] However, it has disadvantages in precision, particularly with hand-drawn graphs where scale and resolution limit accuracy, and it cannot directly identify complex roots.[46] As an example, consider sketching $ y = x^2 - 5x + 6 $; the parabola opens upward, and visual estimation of x-intercepts around x=2 and x=3 provides a quick sense of the roots before applying algebraic methods.[47]

Numerical and Practical Considerations

Avoiding Loss of Significance

When computing the roots of a quadratic equation $ ax^2 + bx + c = 0 $ using the quadratic formula, numerical instability arises due to catastrophic cancellation in floating-point arithmetic, particularly when $ |b| $ is much larger than $ \sqrt{|4ac|} $, causing the terms $ -b \pm \sqrt{b^2 - 4ac} $ to be nearly equal in magnitude but opposite in sign for one root.[48] This subtraction of closely valued large numbers leads to a severe loss of significant digits, resulting in inaccurate computation of the smaller root in absolute value, while the larger root remains reliable.[49] The issue is exacerbated when the discriminant $ D = b^2 - 4ac $ is small relative to $ b^2 $, as the square root $ \sqrt{D} $ approximates $ |b| $, amplifying roundoff errors in finite-precision systems like IEEE 754 double precision.[50] To mitigate this loss of significance, an alternative formulation computes the problematic root by rationalizing the numerator: for the root nearer to $ -c/b $, use $ x = \frac{2c}{-b - \operatorname{sign}(b) \sqrt{D}} $, where the sign ensures addition rather than subtraction of large terms.[48] The stable root (farther from zero) is calculated first via the standard formula with the sign that avoids cancellation—specifically, $ x_1 = \frac{-b - \operatorname{sign}(b) \sqrt{D}}{2a} $—and the second root $ x_2 $ follows from the rationalized form $ x_2 = \frac{2c}{ -b - \operatorname{sign}(b) \sqrt{D} } $, preserving full precision for both.[51] This approach, originally highlighted by Carl Friedrich Gauss and refined by William Kahan, ensures that the computed roots satisfy Vieta's formulas $ x_1 + x_2 = -b/a $ and $ x_1 x_2 = c/a $ within machine epsilon.[49][50] Consider the equation $ 0.0001x^2 + 100000x + 0.002 = 0 $ in double-precision arithmetic (approximately 15 decimal digits). The exact roots are approximately $ -10^9 $ and $ -2 \times 10^{-8} $. Using the standard formula for the smaller root yields $ x \approx 0 $ due to cancellation in $ -b + \sqrt{D} \approx -4 \times 10^{-12} $, losing all precision beyond the first few digits.[49] Applying the rationalized form gives $ x \approx -2 \times 10^{-8} $, accurate to nearly full precision, while the larger root remains stable in both methods.[48] For software implementations, normalize the equation to reduced (monic) form by dividing coefficients by $ a $ before applying the stable algorithm, which scales the problem to unit leading coefficient and reduces overflow risks.[51] Additionally, compute the discriminant with extra precision—such as Kahan's method of representing $ b^2 $ and $ 4ac $ as sums of high- and low-order parts to avoid underflow in $ D $—ensuring robustness across floating-point environments like MATLAB or C++.[50] Many numerical libraries incorporate these techniques to guarantee backward stability.[51]

Reduced Form and Vieta's Formulas

The reduced form of a quadratic equation, also known as the monic form, is obtained by dividing the general equation ax2+bx+c=0ax^2 + bx + c = 0 (with a≠0a \neq 0) by the leading coefficient aa, yielding x2+px+q=0x^2 + px + q = 0, where p=b/ap = b/a and q=c/aq = c/a.[52] This normalization simplifies the expression by setting the coefficient of x2x^2 to 1, facilitating comparisons and substitutions without altering the roots.[52] Vieta's formulas, named after the French mathematician François Viète (1540–1603), establish relationships between the coefficients of a polynomial and the sums and products of its roots.[53] For the general quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 with roots rr and ss, the sum of the roots is r+s=−b/ar + s = -b/a and the product is rs=c/ars = c/a.[18] In the reduced form x2+px+q=0x^2 + px + q = 0, these relations simplify to r+s=−pr + s = -p and rs=qrs = q.[18] These formulas arise from expanding the factored representation of the equation.[18] Assuming roots rr and ss, the monic quadratic can be written as (x−r)(x−s)=0(x - r)(x - s) = 0, which expands to
x2−(r+s)x+rs=0. x^2 - (r + s)x + rs = 0.
Comparing coefficients with x2+px+q=0x^2 + px + q = 0 gives p=−(r+s)p = -(r + s) and q=rsq = rs, directly yielding the sum and product relations.[18] For the general case, scaling by aa preserves the root symmetries.[18] Vieta's formulas enable the construction of quadratic equations directly from known roots, which is particularly useful for substitutions in solving higher-degree equations or analyzing symmetric properties.[53] For instance, if the roots are 2 and 3, then the sum is 5 and the product is 6, so the reduced equation is x2−5x+6=0x^2 - 5x + 6 = 0.[18] Conversely, for the equation x2+4x−5=0x^2 + 4x - 5 = 0, Vieta's formulas confirm that the roots satisfy r+s=−4r + s = -4 and rs=−5rs = -5, allowing verification without explicit solving.[18] Vieta's formulas are frequently applied in educational settings, including CBSE Class 10 Mathematics board examinations and sample papers, where questions often require direct use of the sum and product of roots to solve problems or compute related expressions. Examples from recent CBSE materials include:
  • For the equation 2x2−9x+4=02x^2 - 9x + 4 = 0, the sum of the roots is 9/29/2 and the product is 22.
  • The equation x2+3x−10=0x^2 + 3x - 10 = 0 has roots 22 and −5-5, with sum −3-3 and product −10-10.
  • The quadratic equation with roots 22 and −5-5 is x2+3x−10=0x^2 + 3x - 10 = 0.
  • For the quadratic polynomial 5x2+5x+1=05x^2 + 5x + 1 = 0, if α\alpha and β\beta are the roots, then α2+β2=(α+β)2−2αβ=(−1)2−2(1/5)=1−2/5=3/5\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = (-1)^2 - 2(1/5) = 1 - 2/5 = 3/5.[54]

Applications and Examples

Quadratic equations are applied in numerous real-world contexts, modeling parabolic trajectories in physics and optimizing quadratic functions in engineering, business, and other fields. Recent educational resources and lesson plans from 2025–2026 highlight these applications in sports, business optimization, civil and mechanical engineering, and education.[55][56]

Real-World Physical Examples

Quadratic equations frequently model projectile motion in physics, where the height of an object follows a parabolic trajectory under constant acceleration due to gravity. Specific examples include basketball shots (such as slam dunks), fireworks trajectories, and water paths in fountains. The vertical position $ h(t) $ as a function of time $ t $ is given by
h(t)=h0+v0t−12gt2, h(t) = h_0 + v_0 t - \frac{1}{2} g t^2,
where $ h_0 $ is the initial height, $ v_0 $ is the initial vertical velocity, and $ g $ is the acceleration due to gravity (approximately 9.8 m/s² in SI units or 32 ft/s² in US customary units, yielding -4.9 t² or -16 t² respectively for the quadratic term). To find the time of flight when the object returns to the initial height ($ h(t) = h_0 $), set $ h(t) = h_0 $, simplifying to $ 0 = v_0 t - \frac{1}{2} g t^2 $, or $ t (v_0 - \frac{1}{2} g t) = 0 $, with solutions $ t = 0 $ and $ t = \frac{2 v_0}{g} $. The discriminant of this quadratic, $ D = v_0^2 $, is always non-negative for real $ v_0 $, ensuring feasible solutions, but for queries like maximum reachable height, a negative discriminant indicates physical impossibility (e.g., no real time to exceed the vertex height).[57][58] In engineering design, quadratic equations describe parabolic shapes that optimize structural and functional performance. Applications include the arches of bridges for load distribution, the reflective surfaces of satellite dishes to focus signals at a focal point, the curved profiles of certain dams for pressure management, and the contours of ramps for smooth transitions. These parabolic forms, derived from quadratic functions, enhance stability, efficiency, and safety in civil and mechanical engineering.[55] In engineering optimization, quadratic equations arise when maximizing the area of a rectangle given a fixed perimeter, such as in fencing or material allocation problems. For a perimeter $ P = 2(l + w) $, where $ l $ is length and $ w $ is width, express width as $ w = \frac{P}{2} - l $; the area is then $ A(l) = l \left( \frac{P}{2} - l \right) = -\ l^2 + \frac{P}{2} l $. The maximum area occurs at the vertex $ l = -\frac{b}{2a} = \frac{P}{4} $, yielding $ w = l $ (a square) with $ A_{\max} = \left( \frac{P}{4} \right)^2 $. For $ P = 100 $ yards, dimensions are 25 yards by 25 yards, area 625 square yards; the positive discriminant $ D = \left( \frac{P}{2} \right)^2 > 0 $ confirms two real roots bounding the feasible domain. Units of area (e.g., square meters) interpret the practical enclosure size.[59][60] In electrical engineering, quadratic equations describe the behavior of RLC circuits, particularly for determining resonance frequencies in series or parallel configurations. The governing differential equation for charge $ q(t) $ is $ L \frac{d^2 q}{dt^2} + R \frac{d q}{dt} + \frac{1}{C} q = 0 $, leading to the characteristic quadratic $ s^2 + \frac{R}{L} s + \frac{1}{LC} = 0 $. The roots are $ s = -\frac{R}{2L} \pm \sqrt{ \left( \frac{R}{2L} \right)^2 - \frac{1}{LC} } $, where the discriminant $ D = \left( \frac{R}{2L} \right)^2 - \frac{1}{LC} $ determines damping: underdamped ($ D < 0 $) yields oscillatory resonance at angular frequency $ \omega = \sqrt{ \frac{1}{LC} - \left( \frac{R}{2L} \right)^2 } $ in radians per second, feasible for real circuit oscillations. For example, with $ L = 33.43 $ mH, $ C = 1 $ μF, and low $ R $, resonance occurs near 862 Hz.[61][62][63]

Mathematical and Scientific Applications

In calculus, quadratic functions serve as a cornerstone for optimization problems, particularly in identifying maxima and minima through the vertex of the parabola. The vertex of a quadratic function $ f(x) = ax^2 + bx + c $ occurs at $ x = -\frac{b}{2a} $, providing the input value where the function achieves its extreme value; if $ a > 0 $, this is a minimum, and if $ a < 0 $, it is a maximum. For instance, consider maximizing the area $ A $ of a rectangular garden enclosed by 40 meters of fencing on three sides, with the fourth side along a river. Let the width perpendicular to the river be $ x $ meters (two sides); then the length parallel to the river is $ y = 40 - 2x $ meters, yielding $ A(x) = x(40 - 2x) = -2x^2 + 40x $. The vertex at $ x = -\frac{40}{2(-2)} = 10 $ gives dimensions of 10 m by 20 m, maximizing the area at $ A(10) = 200 $ square meters. This approach highlights how quadratics enable precise determination of optimal configurations in constrained scenarios. In economics, quadratic models are widely used to represent profit functions and pricing optimization strategies, where revenue typically increases with quantity sold or adjusts with price but eventually diminishes due to market saturation, resulting in a downward-opening parabola. The profit $ P(x) $ for producing $ x $ units is given by $ P(x) = R(x) - C(x) $, often simplifying to a quadratic form $ P(x) = ax^2 + bx + c $ with $ a < 0 $, whose vertex indicates the production level or pricing point maximizing profit or revenue. For example, suppose a firm's profit function is $ P(x) = -2x^2 + 100x - 500 $, where $ x $ is the number of units produced. The maximum occurs at $ x = -\frac{100}{2(-2)} = 25 $ units, yielding $ P(25) = -2(25)^2 + 100(25) - 500 = 750 $ dollars. This interpretation allows economists to advise on optimal output levels or pricing to avoid overproduction losses.[8] Quadratic equations appear in biology, particularly in population genetics, to model mean fitness as a function of allele frequencies, capturing stabilizing selection where intermediate frequencies yield optimal population viability. Under Hardy-Weinberg equilibrium, for a locus with alleles A (frequency $ p $) and S (frequency $ q = 1 - p $), the mean fitness $ \bar{w}(p) $ is $ \bar{w}(p) = w_{AA} p^2 + 2 w_{AS} p q + w_{SS} q^2 $, a quadratic in $ p $. In the case of sickle cell anemia, with fitness values $ w_{AA} = 0.76 $, $ w_{AS} = 1.0 $, and $ w_{SS} = 0.20 $, the function becomes $ \bar{w}(p) = 0.76p^2 + 2p(1-p) + 0.20(1-p)^2 $. Simplifying, $ \bar{w}(p) = -1.04p^2 + 1.6p + 0.20 $; the maximum at $ p = -\frac{1.6}{2(-1.04)} \approx 0.77 $ indicates the allele frequency balancing malaria resistance and anemia risks for highest population fitness. This model underscores how quadratics reveal evolutionary equilibria.[64] In geometry, quadratic equations arise when applying the Pythagorean theorem to right triangles with variable sides, leading to equations that must be solved for unknown lengths, and in area problems where dimensions are related quadratically. For example, consider a right triangle where one leg is $ x $ units, the other leg is $ x + 7 $ units, and the hypotenuse is $ x + 8 $ units. The Pythagorean theorem gives $ x^2 + (x + 7)^2 = (x + 8)^2 $, expanding to $ x^2 + x^2 + 14x + 49 = x^2 + 16x + 64 $, or $ x^2 - 2x - 15 = 0 $. Solving via the quadratic formula, $ x = \frac{2 \pm \sqrt{4 + 60}}{2} = \frac{2 \pm 8}{2} $, yields $ x = 5 $ or $ x = -3 $ (discarding the negative), so the sides are 5, 12, and 13 units. This extends the theorem to dynamic configurations, such as ladder problems against walls. Similarly, for areas, quadratics model trade-offs in enclosure design, as in the fencing example above, emphasizing symmetry and extrema for maximal spatial efficiency.[65]

Historical Development

Ancient and Medieval Solutions

The earliest known solutions to quadratic equations emerged in ancient Mesopotamia around 2000 BCE, where Babylonian mathematicians employed geometric methods to address problems involving areas and dimensions of fields or structures. These solutions were recorded on cuneiform clay tablets and typically involved iterative numerical procedures or "cut-and-paste" geometric manipulations to find lengths satisfying quadratic relations, such as dividing a rectangular field into squares and rectangles. A prominent artifact is the Yale Babylonian Collection tablet YBC 7289 (c. 1800–1600 BCE), which provides a highly accurate sexagesimal approximation of 2\sqrt{2} as 1;24,51,10 (equivalent to about 1.41421356), derived from solving a quadratic arising in geometric contexts like diagonal calculations.[66][67] In ancient Greece, Euclid formalized geometric approaches to quadratic problems in his Elements (c. 300 BCE), particularly in Book II, where propositions demonstrate algebraic identities through constructions, such as Proposition II.11, which divides a given straight line in extreme and mean ratio (the golden ratio), providing a geometric solution to the associated quadratic equation, and Proposition II.14, which constructs a square equal to a given rectilinear figure, effectively solving quadratic equations of the form x2+px=qx^2 + px = q via a geometric completion of the square, transforming rectangles into squares without explicit numerical computation. This method built on earlier Pythagorean traditions and emphasized deductive proofs, treating quadratics as applications of area equivalences rather than abstract equations. Euclid's work preserved and systematized these techniques, influencing subsequent European mathematics.[68][69][67] Indian mathematicians advanced quadratic solutions algebraically in the early medieval period, with Brahmagupta's Brahmasphuṭasiddhānta (628 CE) providing verbal rules for extracting positive roots from equations like ax2+bx=cax^2 + bx = c. Brahmagupta described a procedure akin to the quadratic formula, stating: "Put down twice the square root of a given square multiplied by a multiplier and increased or diminished by an arbitrary number," which generates solutions iteratively while restricting to positive values and rational coefficients, often in astronomical or inheritance contexts. This represented a shift toward symbolic manipulation, though still tied to practical problems, and excluded negative or irrational roots as non-physical.[70][67] In the Islamic Golden Age, Muhammad ibn Musa al-Khwarizmi's Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala (c. 820 CE) introduced a systematic algebraic framework for quadratics, classifying them into six types based on coefficients (e.g., "squares equal to roots" or "squares plus roots equal numbers") and solving via completion of the square with geometric justifications. Al-Khwarizmi avoided negative roots, interpreting them as impossible in real-world applications like commerce or land measurement, and provided step-by-step rhetorical algorithms without symbols. His treatise, preserved through Arabic manuscripts and later Latin translations—such as Robert of Chester's 1145 version and Gerard of Cremona's 12th-century rendering—facilitated the transmission of these methods to medieval Europe, bridging ancient geometric traditions with emerging algebraic practices.[71][67]

Modern Formulation

The Renaissance marked a pivotal shift in the treatment of quadratic equations, building on earlier geometric foundations to emphasize algebraic methods. In 1545, Gerolamo Cardano published Ars Magna, which, while primarily focused on solving cubic and quartic equations, included systematic approaches to quadratics as special cases, presenting solutions in radical form and introducing the use of complex quantities to handle intermediate steps.[72] Lodovico Ferrari, Cardano's associate, contributed to the same work by developing a general method for quartics that relied on resolving associated cubics, indirectly refining quadratic techniques through the emphasis on polynomial depression and substitution.[73] These advancements promoted a more analytic perspective, standardizing symbolic manipulation over purely geometric constructions. François Viète, in the late 16th century, further revolutionized the field by introducing systematic symbolic notation in his 1591 work In artem analyticam isagoge, using letters to represent both unknowns and parameters, which enabled the expression of quadratic relations in a general form.[74] Viète's formulas, articulating the sum and product of roots in terms of coefficients—for a quadratic ax2+bx+c=0ax^2 + bx + c = 0, the sum of roots equals −b/a-b/a and the product equals c/ac/a—provided a foundational link between roots and coefficients, facilitating derivations without specific numerical values.[74] Shortly thereafter, in 1594, Simon Stevin provided the first general solution to the quadratic equation covering all cases in his work Arithmétique or De Thiende. This symbolic framework laid the groundwork for the modern quadratic formula. In the 17th and 18th centuries, René Descartes integrated quadratic equations with geometry through his 1637 La Géométrie, establishing coordinate systems that represented quadratic relations as conic sections, particularly parabolas, allowing algebraic solutions to be visualized and manipulated geometrically.[75] By the 19th century, Carl Friedrich Gauss advanced the understanding of quadratic roots by proving the fundamental theorem of algebra in multiple versions starting in 1799, demonstrating that every non-constant polynomial, including quadratics, has roots in the complex plane, thus ensuring all solutions could be accounted for using complex numbers.[76] The 20th century saw the quadratic formula adapted for computational contexts, where numerical stability became critical in digital computing due to floating-point arithmetic limitations. Early implementations highlighted issues like catastrophic cancellation in the standard formula when b2≫4acb^2 \gg 4ac, leading to loss of precision in one root; solutions involved recomputing the smaller root using Vieta's product relation to avoid subtraction of close values.[48] These refinements, developed amid the rise of electronic computers, ensured reliable evaluation and contributed to the formula's universal acceptance as a staple in scientific computing and engineering applications today.

Advanced Topics

Complex and Trigonometric Solutions

When the discriminant D=b2−4acD = b^2 - 4ac of the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is negative, the roots are complex conjugates of the form −b±i∣D∣2a\frac{-b \pm i \sqrt{|D|}}{2a}.[28] This arises because the square root of a negative number introduces the imaginary unit i=−1i = \sqrt{-1}, ensuring the roots are non-real but come in pairs that are mirror images across the real axis in the complex plane.[77] The conjugate property follows from the fact that the coefficients aa, bb, and cc are real, preserving the equation's reality under conjugation.[28] The complex roots can be expressed in polar form to highlight their magnitude and argument, which is useful for operations like multiplication or finding powers. For a root z=α+βiz = \alpha + \beta i, the magnitude is r=α2+β2r = \sqrt{\alpha^2 + \beta^2} and the argument is θ=tan⁡−1(β/α)\theta = \tan^{-1}(\beta / \alpha) (adjusted for quadrant), yielding z=r(cos⁡θ+isin⁡θ)z = r (\cos \theta + i \sin \theta).[78] For the conjugate pair, the magnitudes are identical, while the arguments are θ\theta and −θ-\theta.[77] This trigonometric representation leverages Euler's formula eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta, facilitating computations in fields like signal processing or electrical engineering where quadratic equations model oscillatory systems.[78] Consider the equation x2+x+1=0x^2 + x + 1 = 0, where a=1a = 1, b=1b = 1, c=1c = 1, and D=1−4=−3<0D = 1 - 4 = -3 < 0. The roots are x=−1±i32x = \frac{-1 \pm i \sqrt{3}}{2}.[28] In polar form, the magnitude r=(−12)2+(32)2=1r = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = 1, with arguments θ=2π3\theta = \frac{2\pi}{3} and θ=−2π3\theta = -\frac{2\pi}{3} (or 120∘120^\circ and −120∘-120^\circ), so the roots are cos⁡2π3+isin⁡2π3\cos\frac{2\pi}{3} + i \sin\frac{2\pi}{3} and cos⁡(−2π3)+isin⁡(−2π3)\cos\left(-\frac{2\pi}{3}\right) + i \sin\left(-\frac{2\pi}{3}\right).[78] These are the non-real cube roots of unity, illustrating how complex quadratic roots connect to roots of higher-degree polynomials.[77] The trigonometric form provides an alternative view of complex solutions, expressing them via cosine and sine functions for the real and imaginary parts relative to the polar angle. Specifically, for roots −b2a±i∣D∣2a\frac{-b}{2a} \pm i \frac{\sqrt{|D|}}{2a}, the form aligns with r(cos⁡θ±isin⁡θ)r \left( \cos \theta \pm i \sin \theta \right), where the ±\pm captures the conjugate pair through symmetric arguments.[78] This representation emphasizes the rotational aspect in the complex plane and is derived from the standard rectangular form without additional identities beyond De Moivre's theorem.[77] For cases with positive discriminant (D>0D > 0), where roots are real and distinct, hyperbolic functions offer an alternative expression to enhance numerical stability, particularly when ∣b∣|b| is large compared to D\sqrt{D}. The roots can be reformulated using inverse hyperbolic functions, such as relating cosh⁡−1\cosh^{-1} solutions to quadratics via ln⁡(x+x2−1)\ln\left(x + \sqrt{x^2 - 1}\right) for x≥1x \geq 1, avoiding cancellation errors in direct computation.[79] This approach is valuable in computational contexts like solving boundary value problems or optimizing algorithms where precision matters.[79]

Generalizations to Other Contexts

Quadratic equations extend naturally to other algebraic structures beyond the real or complex numbers, such as finite fields, where the characteristic of the field influences the solvability and solution methods. In fields of characteristic not equal to 2, the standard quadratic formula applies, but in characteristic 2, the equation x2+bx+c=0x^2 + bx + c = 0 requires alternative approaches because the discriminant b2−4acb^2 - 4ac simplifies to b2b^2 (since 4=04 = 0) and division by 2 is impossible. If b=0b = 0, the equation reduces to x2=cx^2 = c, solvable by finding square roots in the field, which exist for perfect fields like finite fields of characteristic 2. If b≠0b \neq 0, a substitution y=x/by = x/b transforms it to y2+y+(c/b2)=0y^2 + y + (c/b^2) = 0, and solutions depend on the trace function: the equation has solutions if the absolute trace of c/b2c/b^2 is zero, with explicit formulas involving half-traces or iterative methods in extensions like GF(2m)\mathrm{GF}(2^m).[80] In the context of quadratic forms, the homogeneous equation ax2+bxy+cy2=0ax^2 + bxy + cy^2 = 0 in two variables over a field represents the projective line at infinity or degenerate conic sections when set to zero, but more generally, the full quadratic form ax2+bxy+cy2+dx+ey+f=0ax^2 + bxy + cy^2 + dx + ey + f = 0 defines conic sections such as ellipses, parabolas, or hyperbolas in the real plane, classified by the discriminant b2−4acb^2 - 4ac of the quadratic part. Over other fields, these forms retain their bilinear structure, with the associated symmetric matrix (ab/2b/2c)\begin{pmatrix} a & b/2 \\ b/2 & c \end{pmatrix} determining isotropy or anisotropy via eigenvalues or Hasse invariants. In projective geometry, such forms classify conics up to coordinate changes, linking to broader quadratic hypersurface theory.[81] For linear algebra over any field, the characteristic equation of a 2×22 \times 2 matrix A=(pqrs)A = \begin{pmatrix} p & q \\ r & s \end{pmatrix} is det⁡(A−λI)=λ2−(p+s)λ+(ps−qr)=0\det(A - \lambda I) = \lambda^2 - (p+s)\lambda + (ps - qr) = 0, a monic quadratic polynomial whose roots are the eigenvalues, enabling spectral decomposition when solvable. This quadratic form arises directly from the determinant expansion and generalizes to higher dimensions, but for n=2n=2, it mirrors the scalar case with trace and determinant as coefficients. Solutions via the quadratic formula hold in characteristic not 2, while in characteristic 2, eigenvalue computation adapts similarly to field-specific methods.[82] In multivariable settings, quadratic equations generalize to hypersurfaces defined by ∑i,jaijxixj+∑ibixi+c=0\sum_{i,j} a_{ij} x_i x_j + \sum_i b_i x_i + c = 0 in nn-dimensional space, forming quadric hypersurfaces that extend conic sections to higher dimensions, such as ellipsoids or hyperboloids in R3\mathbb{R}^3 and their projective analogs. These are algebraic varieties of degree 2, classified by the signature of the quadratic form matrix or rank, with applications in optimization and geometry where the Hessian captures second-order behavior. Over finite fields, such hypersurfaces count points via zeta functions, influencing enumerative geometry./12:_Vectors_in_Space/12.06:_Quadric_Surfaces) A concrete example occurs in the finite field GF(2)\mathrm{GF}(2), the field with two elements {0,1}\{0,1\} where 1+1=01+1=0. The quadratic x2+x+a=0x^2 + x + a = 0 (noting x2+x=0x^2 + x = 0 for all elements) has roots x=0x=0 and x=1x=1 when a=0a=0, and no solutions when a=1a=1, actually illustrating that non-trivial quadratics may lack roots in the field. Due to the finiteness of the field, solutions can be verified by checking all elements exhaustively. This ties to coding theory, where quadratic residues over GF(2m)\mathrm{GF}(2^m) decode Reed-Solomon codes via solving x2+bx+c=0x^2 + bx + c = 0 for error locations, using half-trace formulas for efficiency, as in Berlekamp's algorithm variants; similarly, nonlinear codes like quadratic residue codes over GF(2)\mathrm{GF}(2) use such forms for parity-check matrices, achieving optimal distance properties.[83]

References

  1. [PDF] ▫ Introduction to Quadratic Equations. Definition of a quadratic ...
  2. Algebra - Quadratic Equations - Part I - Pauls Online Math Notes
  3. BioMath: Quadratic Functions
  4. Quadratic, cubic and quartic equations - MacTutor
  5. Completing the Square: The prehistory of the quadratic formula
  6. Algebra - Quadratic Equations : A Summary - Pauls Online Math Notes
  7. [PDF] Quadratic Functions and Equations
  8. Applications of Quadratic Equations - Pauls Online Math Notes
  9. [PDF] A Real World Example of Solving a Quadratic Equation in Movie CGI
  10. Quadratic equation | Solutions, Roots & Coefficients - Britannica
  11. 11.4: Solve Quadratic Equations Using the Quadratic Formula
  12. Question Corner -- The Origin Of The Word Quadratic
  13. 3.2: Quadratic Functions - Mathematics LibreTexts
  14. Vertex form introduction (video) | Khan Academy
  15. None
  16. Discriminant Formula, Rules & Solutions - Lesson | Study.com
  17. Vieta's Formula | Brilliant Math & Science Wiki
  18. Quadratic Equations - Department of Mathematics at UTSA
  19. ac Method of Factoring
  20. Quadratic Equations - College Algebra - West Texas A&M University
  21. [PDF] Completing the square | Mathcentre
  22. Completing the Square | Mathematical Association of America
  23. Vertex Form of Quadratic Equation - MathBitsNotebook(A1)
  24. Algebra - Quadratic Equations - Part II - Pauls Online Math Notes
  25. Quadratic Formula -- from Wolfram MathWorld
  26. Polynomials and Factoring - Worked Examples
  27. Parabolas - Algebra - Pauls Online Math Notes
  28. MFG The Vertex of a Parabola
  29. [PDF] Section 4.1 Quadratic Functions - Math@LSU
  30. Euclid's Elements, Book II, Proposition 6 - Clark University
  31. [PDF] 3.7. Geometric Solution of Quadratic Equations
  32. Carlyle Circle -- from Wolfram MathWorld
  33. [PDF] Constructible numbers: Euclid and beyond - CSUSB ScholarWorks
  34. 3-03 Solve Quadratic Equations by Graphing and Finding Square ...
  35. [PDF] Ways To Solve Quadratic Equations
  36. Graphs of Quadratic Functions - West Texas A&M University
  37. [PDF] Using Desmos to Support Conceptual Understanding of Quadratic ...
  38. [PDF] 3 Quadratic Functions Big Ideas Learning
  39. [PDF] 4 Ways To Solve Quadratic Equations
  40. [PDF] Lesson 9 – Solving Quadratic Equations
  41. [PDF] Numerically Stable Method for Solving Quadratic Equations
  42. [PDF] Approximating solutions to the quadratic equation
  43. http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf
  44. [PDF] The Ins and Outs of Solving Quadratic Equations with Floating-Point ...
  45. Quadratic Equations - Australian Mathematical Sciences Institute
  46. [PDF] françois viète and his contribution to mathematics - arXiv
  47. [PDF] Quadratic Applications - Gordon State College
  48. 3.4 Projectile Motion – College Physics - UCF Pressbooks
  49. [PDF] Quadratic Functions and Their Graphs
  50. 4.4 Applied Optimization
  51. None
  52. None
  53. [PDF] What Is The Discriminant In A Quadratic Equation
  54. [PDF] Applications of Max & Min I Chapter 6: Quadratic Equations Page 1 ...
  55. [PDF] Applications of Linear and Quadratic Functions in Business and ...
  56. Quadratic Function Applications
  57. Quadratic Applications - MathBitsNotebook(A1)
  58. How the estimate of 2 on YBC 7289 may have been calculated
  59. [PDF] A brief history of quadratic equations for mathematics educators
  60. Euclid's Elements, Book II, Proposition 14 - Clark University
  61. [PDF] Equation Solving in Indian Mathematics - DiVA portal
  62. [PDF] A Brief History of Quadratic Equations - DiVA portal
  63. Girolamo Cardano - Biography - MacTutor
  64. Lodovico Ferrari (1522 - 1565) - Biography - MacTutor
  65. François Viète - Biography - MacTutor - University of St Andrews
  66. René Descartes - Biography
  67. Fund theorem of algebra - MacTutor History of Mathematics
  68. [PDF] Understanding Complex Numbers and Identifying Complex Roots ...
  69. 4. Polar and Exponential Forms - Pauls Online Math Notes
  70. [PDF] 5.6 Quadratic and Cubic Equations
  71. [PDF] arXiv:1702.05789v2 [math.HO] 2 Aug 2019
  72. [PDF] Math 4571 (Advanced Linear Algebra)
  73. The Characteristic Polynomial
  74. [PDF] FAST DECODING OF A dmin - 6 RS CODE
  75. Parabola -- from Wolfram MathWorld
  76. Euclid's Elements, Book VI, Proposition 30
  77. Completing the Square - Brilliant Math & Science Wiki
  78. Euclid's Elements, Book II, Proposition 11
  79. Real-Life Applications of Quadratic Equations
  80. Applications of Quadratics | Real-World Applications of Quadratic Equations
  81. How (and When) to Complete the Square: 5 Simple Steps
  82. The Quadratic Formula
  83. Solving Quadratic Equations: Picking a Method
  84. CBSE Class X Mathematics Standard Sample Question Paper 2023-24
User Avatar
No comments yet.