## 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.
$x^2+4x+3=0 \\[6pt]a=1,\hspace{6 pt} b=4,\hspace{6 pt} c=3$ \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}
$3x^2+7x-2=0 \\[6pt]a=3,\hspace{6 pt} b=7,\hspace{6 pt} c=-2$ \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}