# The quadratic formula

## In a nutshell

A quadratic equation in the form $ax^2+bx+c=0$ can be solved using the quadratic formula. Sometimes, it is necessary to use the quadratic formula to solve a quadratic rather than factorising, as not all quadratics can be factorised easily.

## The quadratic formula

The quadratic formula is

$x= \dfrac{-b \pm \sqrt{b^2-4ac}} {2a}$

Where $a,b$ and $c$ are the co-efficients from the quadratic equation in the form

$ax^2+bx+c=0$

Substitute the values of $a, b$ and $c$ from the quadratic equation into the quadratic formula, and calculate the solutions for $x$.

### Example 1

*Use the quadratic formula to solve *

$2x^2+7x+3=0$

*From the quadratic equation,* $a=2, b=7$ *and* $c=3$. Substitute the numbers into the quadratic formula.

$x= \dfrac{-b \pm \sqrt{b^2-4ac}} {2a}$

*$x= \dfrac {-7 \pm \sqrt {7^2-4\times2\times3}} {2 \times 2}\\$*

*Work out the value inside the square root.*

*$x= \dfrac {-7 \pm \sqrt {25}} {4}$*

*$x= \dfrac {-7 \pm 5} {4}\\$*

*The* $\pm$* symbol means you have to do plus or minus, so there will be two calculations*

*$x = \dfrac {-7+5} 4 \space or \space x= \dfrac {-7-5} 4$*

*Therefore, the answers are*

*$\underline{x=-\frac 1 2} \space or \space \underline{x=-3}$*