A Mathematical Web

Roots of Polynomials

The Quadratic

A polynomial of degree two is called a quadratic, and an equation involving a quadratic polynomial whose value is zero is called a quadratic equation. The solutions of a quadratic equation are called the roots of the polynomial. Figure 1 shows the graph of a quadratic polynomial. The roots of the polynomial are the points where the graph crosses the x-axis.

Figure 1. A quadratic polynomial*

The means of finding the roots of quadratic polynomials were known to the Babylonians and others thousands of years ago. Today, we can write our solutions more elegantly using modern mathematical notation. A general quadratic polynomial can be written in the form

p(x) = ax2 + bx + c.

Using this notation, we can write the roots of p as

( 1/(2a) ) ( -b ± [ b2 - 4ac ]1/2 ).

We can see that whenever b2 - 4ac < 0 then there are two complex roots with non-zero imaginary parts. If b2 - 4ac = 0 then there is exactly one root, which is real. Otherwise, there are two real roots.

* By Original hand-drawn version: N.Mori Updated hand-drawn version: Rubber Duck (☮ • ✍) [Public domain], via Wikimedia Commons