# The Quadratic Formula

## What is the Quadratic Formula?

The quadratic formula is derived from the general form of the a quadratic equation $$ax^2+bx+c=0$$. Using the coefficients $$a$$, $$b$$ and $$c$$, we can find the roots of the equation using: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

## When to use the Quadratic Formula

It will work for any quadratic equation of the form $$ax^2+bx+c=0$$ where $$a\neq0$$. Sometimes completing the square or factorising may be quicker if the answers are simple, but this method can be used regardless.
Click here for more information about when to use each method.

## Worked Example 1: it's easy as plug and play

$x^2+4x+3=0 \\[6pt]a=1,\hspace{6 pt} b=4,\hspace{6 pt} c=3$ \begin{align} x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\[8pt]x&=\frac{-4\pm\sqrt{4^2-4\times1\times3}}{2\times1} \\[8pt]x&=\frac{-4\pm\sqrt{16-12}}{2} \\[8pt]x&=\frac{-4\pm\sqrt{4}}{2} \\[8pt]x&=\frac{-4\pm2}{2} \\[8pt]x&=-2\pm1 \\[8pt]x&=-3\hspace{6 pt}\text{or }-1 \end{align}

## Worked Example 2: not-so-nice values

$3x^2+7x-2=0 \\[6pt]a=3,\hspace{6 pt} b=7,\hspace{6 pt} c=-2$ \begin{align} x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\[8pt]x&=\frac{-7\pm\sqrt{7^2-4\times3\times(-2)}}{2\times3} \\[8pt]x&=\frac{-7\pm\sqrt{49+24}}{6} \\[8pt]x&=\frac{-7\pm\sqrt{73}}{6} \\[8pt]x&=\frac{-7+\sqrt{73}}{6} \hspace{6 pt}\text{or }\frac{-7-\sqrt{73}}{6} \end{align}