Skip to main content Skip to Search Box

Definition: quadratic equation from Philip's Encyclopedia

Algebraic equation in which the highest exponent of the variable is 2; an equation of the second degree. A quadratic equation has the general form ax2 + bx + c = 0, where a, b and c are constants. It has, at most, two solutions (roots), given by the formula x = [-b ± √(b2 - 4ac)]/2a.

Summary Article: quadratic equation from The Hutchinson Unabridged Encyclopedia with Atlas and Weather Guide

In mathematics, a polynomial equation of second degree (that is, an equation containing as its highest power the square of a variable, such as x2). The general formula of such equations is

ax2 + bx + c = 0

in which a, b, and c are real numbers, and only the coefficienta cannot equal 0.

In coordinate geometry, a quadratic function represents a parabola.

Some quadratic equations can be solved by factorization (see factor (algebra)), or the values of x can be found by using the formula for the general solution

x = [−b + √(b2 − 4ac)]/2a or

x = [−b − √(b2 − 4ac)]/2a

Depending on the value of the discriminant, b2 − 4ac, a quadratic equation has two real, two equal, or two complex roots (solutions). When b2 − 4ac > 0, there are two distinct real roots. When b2 − 4ac = 0, there are two equal real roots. When b2 − 4ac < 0, there are two distinct complex roots.

Solving quadratics by factorizing If a quadratic equation can be factorized, it can be solved. For example, to solve x2 + x − 6 = 0, the equation should be factorized to give:

(x + 3)(x − 2) = 0

There are now two brackets multiplied together to give 0. This means that one of them must be 0. So either x + 3 = 0 or x − 2 = 0. Therefore, x = −3 or x = 2.

Drawing the graphs of quadratics Quadratic equations always plot parabolas, whereas linear equations plot straight lines. For example, to draw the graph of y = x2 − 2x − 4, the table should be completed:

and then the graph plotted:

From the graph, the solution to the equation y = x2 − 2x − 4 is

x = −1.2 or x = 3.2

Completing the square can solve other quadratics.


Multiplying out brackets and factorizing

Out for the Count – Problem Solving Using Quadratic Equations

Algebra: simultaneous equations

Factorizing quadratic equations

Simplification of surds


Quadratic, Cubic, and Quartic Equations

SOS Mathematics: Algebra

© RM, 2016. All rights reserved.

Related Credo Articles

Full text Article quadratic equation
The Macmillan Encyclopedia

An algebraic equation in which the greatest power of the variable is two. It is usually written in the form ax 2 + bx + c = 0, in...

The SAGE Encyclopedia of Social Science Research Methods

The quadratic equation is an elementary algebraic method for solving equations where the variable of interest is squared. This is a useful...

Full text Article Bombelli, Raffaele (c. 1526–1573)
The Hutchinson Unabridged Encyclopedia with Atlas and Weather Guide

Italian mathematician. An engineer in the service of the bishop of Melfi, he published his major work, his three-volume L'algebra/Algebra, in 1572.

See more from Credo