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

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