# What is the quadratic formula and how is it derived?

Oct 25, 2015

For any general quadratic equation of the form $a {x}^{2} + b x + c = 0$, we have the quadratic formula to find the values of x satisfying the equation and is given by
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

To derive this formula, we use completing the square in the general equation $a {x}^{2} + b x + c = 0$

Dividing throughout by a we get : ${x}^{2} + \frac{b}{a} x + \frac{c}{a} = 0$

Now take the coefficient of x, half it, square it, and add it to both sides and rearrange to get

${x}^{2} + \frac{b}{a} x + {\left(\frac{b}{2 a}\right)}^{2} = {b}^{2} / {\left(4 a\right)}^{2} - \frac{c}{a}$

Now right the left hand side as a perfect square and simplify the right hand side.

$\therefore {\left(x + \frac{b}{2 a}\right)}^{2} = \frac{{b}^{2} - 4 a c}{4 {a}^{2}}$

Now taking the square root on both sides yields :

$x + \frac{b}{2 a} = \pm \frac{\sqrt{\left({b}^{2} - 4 a c\right)}}{2 a}$

Finally solving for x gives

$x = - \frac{b}{2 a} \pm \frac{\sqrt{{b}^{2} - 4 a c}}{2 a}$

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$