1. Quadratic equation
From Wikipedia, the free encyclopedia
This article is about single-variable quadratic equations and their solutions. For more general
information about the single-variable case, see Quadratic function. For the case of more than one
variable, see Quadratic form.
The quadratic formula for the roots of the general quadratic equation
In elementary algebra, a quadratic equation (from the Latin quadratus for "square") is any equation
having the form
where x represents an unknown, and a, b, and c represent known numbers such that a is not
equal to 0. If a = 0, then the equation islinear, not quadratic. The numbers a, b, and c are
the coefficients of the equation, and may be distinguished by calling them, respectively,
the quadratic coefficient, the linear coefficient and the constant or free term.[1]
Because the quadratic equation involves only one unknown, it is called "univariate". The
quadratic equation only contains powers of x that are non-negative integers, and therefore it is
a polynomial equation, and in particular it is a second degree polynomial equation since the
greatest power is two.
Quadratic equations can be solved by a process known in American English as factoring and in
other varieties of English as factorising, by completing the square, by using thequadratic
formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known
as early as 2000 BC.
Solving the quadratic equation[edit]
Figure 1. Plots of quadratic functiony = ax2
+ bx + c, varying each coefficient separately while the other
coefficients are fixed (at values a = 1,b = 0, c = 0)

2. 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.
Factoringby 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.[2]: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"[3]
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.[2]:207
Completing the square[edit]
Main article: Completing the square
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 equationx2 − x − 2 = 0.
The process of completing the square makes use of the algebraic identity

3. which represents a well-defined algorithm that can be used to solve any quadratic
equation.[2]: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. Rearrange the equation so that the constant term c/a is on the right side.
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 the two linear equations.
We illustrate use of this algorithm by solving 2x2 + 4x − 4 = 0
The plus-minus symbol "±" indicates that both x = −1 + √3 and x = −1
− √3 are solutions of the quadratic equation.[4]
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.[5]
The mathematical proof will now be briefly summarized.[6]
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:

4. Some sources, particularly older ones, use alternative parameterizations of the quadratic equation
such as ax2 + 2bx + c = 0 or ax2 − 2bx + c = 0 ,[7]
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.
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:[8]
where p = b/a and q = c/a. This monic 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. 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:[9]

5. 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.
 If the discriminant is negative, then there are no real roots. Rather,
there are two distinct (non-real) complex roots[10]
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.
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

6. In the special case b2 = 4acwhere the
quadratic has only one distinct root (i.e. the
discriminant is zero), the quadratic polynomial
can be factored as
Graphing for real roots[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.
For most of the 20th century, graphing was rarely mentioned as a method for solving quadratic
equations in high school or college algebra texts. Students learned to solve quadratic equations by
factoring, completing the square, and applying the quadratic formula. Recently, graphing calculators
have become common in schools and graphical methods have started to appear in textbooks, but
they are generally not highly emphasized.[12]
Being able to use a graphing calculator to solve a quadratic equation requires the ability to produce
a graph of y = f(x), the ability to scale the graph appropriately to the dimensions of the graphing
surface, and the recognition that when f(x) = 0, x is a solution to the equation. The skills required to
solve a quadratic equation on a calculator are in fact applicable to finding the real roots of any
arbitrary function.
Since an arbitrary function may cross the x-axis at multiple points, graphing calculators generally
require one to identify the desired root by positioning a cursor at a "guessed" value for the root.
(Some graphing calculators require bracketing the root on both sides of the zero.) The calculator
then proceeds, by an iterative algorithm, to refine the estimated position of the root to the limit of
calculator accuracy.