Quadratic equation

This article is about quadratic equations and their solutions. For more general information about quadratic functions, see Quadratic function. For more information about quadratic polynomials, see Quadratic polynomial.

In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is

$ax^{2}+bx+c=0,\,$

where x represents a variable, and a, b, and c, represent coefficients and constants, with a ≠ 0. (If a = 0, the equation becomes a linear equation.)

The constants a, b, and c, are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. Quadratic comes from quadratus, which is the Latin word for "square." Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula. One common use of quadratic equations is computing trajectories in projectile motion.

Plots of real-valued quadratic function ax² + bx + c, varying each coefficient separately

Quadratic formula

A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real. The roots are given by the quadratic formula:

${\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},$

where the symbol "±" indicates that both

${\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}$

and

${\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}$

are solutions.

Discriminant

Example discriminant signs
■ <0: x²+¹⁄₂
■ =0: −⁴⁄₃x²+⁴⁄₃x−¹⁄₃
■ >0: ³⁄₂x²+¹⁄₂x−⁴⁄₃

In the above formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case Greek Delta:

$\Delta =b^{2}-4ac.\,$

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, there are two distinct roots, both of which are real numbers.

For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers—in other cases they may be quadratic irrationals.

• If the discriminant is zero, there is exactly one distinct real root, sometimes called a double root:

$x=-{\frac {b}{2a}}.\,\!$

• If the discriminant is negative, there are no real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:

${\frac {-b}{2a}}+i{\frac {\sqrt {4ac-b^{2}}}{2a}}$

and

${\frac {-b}{2a}}-i{\frac {\sqrt {4ac-b^{2}}}{2a}},$

where 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.

Geometry

For the quadratic function:
f (x) = x² − x − 2 = (x + 1)(x − 2) of a real variable x, the x-coordinates of the points where the graph intersects the x-axis, x = −1 and x = 2, are the roots of the quadratic equation: x² − x − 2 = 0.

The solutions to the quadratic equation

$ax^{2}+bx+c=0,\,$

are also the zeros of the quadratic function:

$f(x)=ax^{2}+bx+c,\,$

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 zeros of f are exactly the x-coordinates of the points where the graph touches the x-axis.

It follows from the above that, 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

The term

$x-r\,$

is a factor of the polynomial

$ax^{2}+bx+c,\$

if and only if r is a root of the quadratic equation

$ax^{2}+bx+c=0.\$

It follows from the quadratic formula that

$ax^{2}+bx+c=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right).$

In the special case ($b^{2}=4ac$) where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as

$ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}.\,\!$

Application to higher-degree equations

Certain higher-degree equations can be brought into quadratic form and solved that way. For example, the 6th-degree equation in x:

$x^{6}-4x^{3}+8=0\,$

can be rewritten as:

$(x^{3})^{2}-4(x^{3})+8=0\,,$

or, equivalently, as a quadratic equation in a new variable u:

$u^{2}-4u+8=0,\,$

where

$u=x^{3}.\,$

Solving the quadratic equation for u results in the two solutions:

$u=2\pm 2i\,.$

Thus

$x^{3}=2\pm 2i\,.$

Concentrating on finding the three cube roots of 2 + 2i – the other three solutions for x will be their complex conjugates – rewriting the right-hand side using Euler's formula:

$x^{3}=2^{\tfrac {3}{2}}e^{{\tfrac {1}{4}}\pi i}=2^{\tfrac {3}{2}}e^{{\tfrac {8k+1}{4}}\pi i}\,$

(since e^2kπi = 1), gives the three solutions:

$x=2^{\tfrac {1}{2}}e^{{\tfrac {8k+1}{12}}\pi i}\,,~k=0,1,2\,.$

Using Eulers' formula again together with trigonometric identities such as cos(π/12) = (√2 + √6) / 4, and adding the complex conjugates, gives the complete collection of solutions as:

$x_{1,2}=-1\pm i,\,$

$x_{3,4}={\frac {1+{\sqrt {3}}}{2}}\pm {\frac {1-{\sqrt {3}}}{2}}i\,$

and

$x_{5,6}={\frac {1-{\sqrt {3}}}{2}}\pm {\frac {1+{\sqrt {3}}}{2}}i.\,$

History

The Babylonians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve a pair of simultaneous equations of the form:

$x+y=p,\ \ xy=q\$

which are equivalent to the equation:

$\ x^{2}+q=px$

The original pair of equations were solved as follows:

1. Form ${\frac {x+y}{2}}$
2. Form $\left({\frac {x+y}{2}}\right)^{2}$
3. Form $\left({\frac {x+y}{2}}\right)^{2}-xy$
4. Form ${\sqrt {\left({\frac {x+y}{2}}\right)^{2}-xy}}={\frac {x-y}{2}}$
5. Find $x,\ y$ by inspection of the values in (1) and (4).

There is evidence pushing this back as far as the Ur III dynasty.

In the Sulba Sutras in ancient India circa 8th century BC quadratic equations of the form ax² = c and ax² + bx = c were explored using geometric methods. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC.

In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation

$\ ax^{2}+bx=c$

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. (Brahmasphutasiddhanta (Colebrook translation, 1817, page 346)

This is equivalent to:

$x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}.$

The Bakhshali Manuscript dated to have been written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y). Mohammad bin Musa Al-kwarismi (Persia, 9th century) developed a set of formulas that worked for positive solutions based on Brahmagupta. The Catalan Jewish mathematician Abraham bar Hiyya Ha-Nasi authored the first book to include the full solution to the general quadratic equation.

The writing of the Chinese mathematician Yang Hui (1238-1298 AD) represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. The first appearance of the general solution in the modern mathematical literature is evidently in an 1896 paper by Henry Heaton.

Derivations

By completing the square

The quadratic formula can be derived by the method of completing the square, so as to make use of the algebraic identity:

$x^{2}+2xh+h^{2}=(x+h)^{2}.\,\!$

Dividing the quadratic equation

$ax^{2}+bx+c=0\,\!$

by a (which is allowed because a is non-zero), gives:

$x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0,\,\!$

or

$x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}$

The quadratic equation is now in a form to which the method of completing the square can be applied. To "complete the square" is to add a constant to both sides of the equation such that the left hand side becomes a complete square:

$x^{2}+{\frac {b}{a}}x+\left({\frac {1}{2}}{\frac {b}{a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {1}{2}}{\frac {b}{a}}\right)^{2},\!$

which produces

$\left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}.\,\!$

The right side can be written as a single fraction, with common denominator 4a². This gives

$\left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.$

Taking the square root of both sides yields

$x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}.$

Isolating x, gives

$x=-{\frac {b}{2a}}\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.$

By Lagrange resolvents

Further information: Lagrange resolvents

An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents, which is an early part of Galois theory. A benefit of this method is that it generalizes to give the solution of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of polynomials of any degree in terms of the symmetry group of their roots, the Galois group.

This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial $x^{2}+px+q,$ and assume that it factors as $x^{2}+px+q=(x-\alpha )(x-\beta ).$ Expanding this out yields

$x^{2}+px+q=x^{2}-(\alpha +\beta )x+\alpha \beta ,$

so $p=-(\alpha +\beta ),q=\alpha \beta .$ Since the order of multiplication does not matter, one can switch α and β and the values of p and q will not change: one says that p and q are symmetric polynomials in α and β. In fact, they are the elementary symmetric polynomials – any symmetric polynomial in α and β can be expressed in terms of $\alpha +\beta$ and αβ. The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree n is related to the ways of rearranging ("permuting") n terms, which is called the symmetric group on n letters, and denoted $S_{n}.$ For the quadratic polynomial, the only way to rearrange 2 terms is to swap them ("transpose" them), and thus solving a quadratic polynomial is simple.

To find the roots α and β, consider their sum and difference:

${\begin{aligned}r_{1}&=\alpha +\beta \\r_{2}&=\alpha -\beta .\\\end{aligned}}$

These are called the Lagrange resolvents of the polynomial; notice that these depend on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations:

${\begin{aligned}\alpha &=\textstyle {\frac {1}{2}}\left(r_{1}+r_{2}\right)\\\beta &=\textstyle {\frac {1}{2}}\left(r_{1}-r_{2}\right).\\\end{aligned}}$

Thus, solving for the resolvents gives the original roots.

Formally, the resolvents are called the discrete Fourier transform (DFT) of order 2, and the transform can be expressed by the matrix $\left({\begin{smallmatrix}1&1\\1&-1\end{smallmatrix}}\right),$ with inverse matrix $\left({\begin{smallmatrix}1/2&1/2\\1/2&-1/2\end{smallmatrix}}\right).$ The transform matrix is also called the DFT matrix or Vandermonde matrix.

Now $r_{1}=\alpha +\beta$ is a symmetric function in α and β, so it can be expressed in terms of p and q, and in fact $r_{1}=-p,$ as noted above. Contrariwise, $r_{2}=\alpha -\beta$ is not symmetric, since switching α and β yields $-r_{2}=\beta -\alpha$ (formally, this is termed a group action of the symmetric group of the roots). Since $r_{2}$ is not symmetric, it cannot be expressed in terms of the roots p and q, as these are symmetric in the roots and thus so is any polynomial expression involving them. However, changing the order of the roots only changes $r_{2}$ by a factor of $-1,$ and thus the square $r_{2}^{2}=(\alpha -\beta )^{2}$ is symmetric in the roots, and thus expressible in terms of p and q. Using the equation $(\alpha -\beta )^{2}=(\alpha +\beta )^{2}-4\alpha \beta$ yields $r_{2}^{2}=p^{2}-4q,$ and thus $r_{2}=\pm {\sqrt {p^{2}-4q}}.$ If one takes the positive root, breaking symmetry, one obtains:

${\begin{aligned}r_{1}&=-p\\r_{2}&={\sqrt {p^{2}-4q}}\\\end{aligned}}$

and thus

${\begin{aligned}\alpha &=\textstyle {\frac {1}{2}}\left(-p+{\sqrt {p^{2}-4q}}\right)\\\beta &=\textstyle {\frac {1}{2}}\left(-p-{\sqrt {p^{2}-4q}}\right)\\\end{aligned}}$

thus the roots are

$\textstyle {\frac {1}{2}}\left(-p\pm {\sqrt {p^{2}-4q}}\right),$

which is the quadratic formula. Substituting $p=b/a,q=c/a$ yields the usual form for when a quadratic is not monic. The resolvents can be recognized as $r_{1}/2=-p/2=-b/2a$ being the vertex, and $r_{2}^{2}=p^{2}-4q$ is the discriminant (of a monic polynomial).

A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the "resolving polynomial") relating $r_{2}$ and $r_{3},$ which one can solve by the quadratic equation, and similarly for a quartic (degree 4) equation, whose resolving polynomial is a cubic, which can in turn be solved. However, the same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and in fact solutions to quintic equations in general cannot be expressed using only roots.

Alternative formula

In some situations it is preferable to express the roots in an alternate form.

$x={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac\ }}}}.$

This alternative requires c to be nonzero; for, if c is zero, the formula correctly gives zero as one root, but fails to give any second, non-zero root. Instead, one of the two choices for ∓ produces a division by zero, which is undefined.

The roots are the same regardless of which expression we use; the alternate form is merely an algebraic variation of the common form:

${\begin{aligned}{\frac {-b+{\sqrt {b^{2}-4ac\ }}}{2a}}&{}={\frac {-b+{\sqrt {b^{2}-4ac\ }}}{2a}}\cdot {\frac {-b-{\sqrt {b^{2}-4ac\ }}}{-b-{\sqrt {b^{2}-4ac\ }}}}\\&{}={\frac {4ac}{2a\left(-b-{\sqrt {b^{2}-4ac}}\right)}}\\&{}={\frac {2c}{-b-{\sqrt {b^{2}-4ac\ }}}}.\end{aligned}}$

The alternative formula can reduce loss of precision in the numerical evaluation of the roots, which may be a problem if one of the roots is much smaller than the other in absolute magnitude. The problem of c possibly being zero can be avoided by using a mixed approach:

$x_{1}={\frac {-b-\operatorname {sgn}(b)\,{\sqrt {b^{2}-4ac}}}{2a}},$

$x_{2}={\frac {c}{ax_{1}}}.$

Here sgn denotes the sign function.

Monic form

Dividing the quadratic equation by a gives the simplified monic form of

$x^{2}+px+q=0$

where p = b⁄a and q = c⁄a. This in turn simplifies the root and discriminant equations somewhat to

$x={\frac {1}{2}}\left(-p\pm {\sqrt {p^{2}-4q}}\,\right)$

$\Delta =p^{2}-4q$

Floating point implementation

A careful floating point computer implementation differs a little from both forms to produce a robust result. Assuming the discriminant, b² − 4ac, is positive and b is nonzero, the code will be something like the following.

$t:=-{\tfrac {1}{2}}{\big (}b+\operatorname {sgn}(b){\sqrt {b^{2}-4ac}}{\big )}\,\!$

$x_{1}:=t/a\,\!$

$x_{2}:=c/t\,\!$

Here sgn(b) is the sign function, where sgn(b) is 1 if b is positive and −1 if b is negative; its use ensures that the quantities added are of the same sign, avoiding catastrophic cancellation. The computation of x₂ uses the fact that the product of the roots is c/a.

See Numerical Recipes in C, Section 5.6: "Quadratic and Cubic Equations".

Viète's formulas

Viète's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form:

$x_{1}+x_{2}=-{\frac {b}{a}}$

and

$x_{1}\cdot x_{2}={\frac {c}{a}}.$

The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex’s x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression:

$x_{V}={\frac {x_{1}+x_{2}}{2}}=-{\frac {b}{2a}}.$

The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving

$y_{V}=-{\frac {b^{2}}{4a}}+c=-{\frac {b^{2}-4ac}{4a}}.$

Generalizations

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

$\pm {\sqrt {b^{2}-4ac}}$

in the formula should be understood as "either of the two elements whose square is

$b^{2}-4ac,\,$

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. Note that 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

In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial

$\displaystyle x^{2}+bx+c$

over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root, so the solution is

$\displaystyle x={\sqrt {c}}$

and note that there is only one root since

$\displaystyle -{\sqrt {c}}=-{\sqrt {c}}+2{\sqrt {c}}={\sqrt {c}}.$

In summary,

$\displaystyle x^{2}+c=(x+{\sqrt {c}})^{2}.$

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

${\frac {b}{a}}R\left({\frac {ac}{b^{2}}}\right)$

and

${\frac {b}{a}}\left(R\left({\frac {ac}{b^{2}}}\right)+1\right).$

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 x + ax + 1 is irreducible over F₄, but 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

• Linear equation
• Cubic function
• Quartic function
• Quintic function
• Fundamental theorem of algebra
• Parabola
• Quadratic function
• Solving quadratic equations with continued fractions
• Periodic points of complex quadratic mappings
• Chakravala method
• Completing the square

References

• Al-Dīnawarī, Abū Ḥanīfa. 820. Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala.
• Stillwell, John, Mathematics and Its History, Springer; 2nd edition (January 27, 2004). ISBN 0-387-95336-1.

External links

• Calculator for solving Quadratics (also solves Cubics and Quartics)
• Weisstein, Eric W. "Quadratic equations". MathWorld.
• 101 uses of a quadratic equation part I Part II
• Quadratic graphical explorer Interactive applet. Sliders for a,b,c show effects on a graph.
• Trigonometric solutions: You Could Learn a Lot from a Quadratic
• Basic Explanation & Application Overview & Factoring
• Graph and Roots of Quadratic Polynomial from cut-the-knot