Quadratic Equations

Computing the roots of a quadratic equation

Quadratic equations are equations of the form

(1) $\begin{equation*} ax^2 + bx + c = 0 \end{equation*}$

where $a \neq 0$ . If $a = 0$ , (1) becomes linear equation, the solution of which is $x=-\frac{c}{b}$ .

A quadratic equation can be solved as follows:

First, calculating $\Delta = b^2 - 4ac$
If $\Delta = 0$ , the equation has one real solution (or root), which is

(2) $\begin{equation*} x = -\frac{b}{2a} \end{equation*}$

If $\Delta > 0$ the equation has two real solutions, $x_{1}$ and $x_{2}$ , where

(3) $\begin{equation*} x_{1}=\frac{-b-\sqrt{\Delta}}{2a} \end{equation*}$

(4) $\begin{equation*} x_{2}=\frac{-b+\sqrt{\Delta}}{2a} \end{equation*}$

If $\Delta < 0$ , then the equation has two complex solutions. To calculating these complex solutions we make use of the complex number $i = \sqrt{-1}$ or $i^2=-1$ , hence $\Delta$ can be written as

(5) $\begin{equation*} \Delta = i^2|\Delta| \end{equation*}$

where $|\Delta|$ is the absolute value of $\Delta$

Substitue (5) into (3) and (4), the two complex solutions of equation (1) are

(6) $\begin{equation*} x_{1}=-\frac{b}{2a}-i\frac{\sqrt{|\Delta|}}{2a} \end{equation*}$

(7) $\begin{equation*} x_{2}=-\frac{b}{2a}+i\frac{\sqrt{|\Delta|}}{2a} \end{equation*}$

which is a pair of conjugate complex numbers

Derivation of (3) and (4) can be found here

A java code for calculating the roots of a quadratic equation is here.

Checking your solutions

A quadratic equation has a form of
aX² + bX + c = 0
To solve your equation, enter the values of a, b, c into the text boxes then click the SUBMIT button

a:	b:	c: