# How do you derive the quadratic formula?

Jan 10, 2015

This is a little bit tricky but also incredibly elegant!
$a {x}^{2} + b x + c = 0$
Take $c$ to the right side:
$a {x}^{2} + b x = - c$
The idea is now to transform the left side in something like ${\left(a + b\right)}^{2}$;
Multiply by $a$;
${a}^{2} {x}^{2} + a b x = - a c$
Multiply by $4$;
$4 {a}^{2} {x}^{2} + 4 a b x = - 4 a c$
Add and subtract ${b}^{2}$ to the left side:
$4 {a}^{2} {x}^{2} + 4 a b x + {b}^{2} - {b}^{2} = - 4 a c$
Take the $- {b}^{2}$ to the right:
$4 {a}^{2} {x}^{2} + 4 a b x + {b}^{2} = {b}^{2} - 4 a c$
The left side can be written as:
${\left(2 a x + b\right)}^{2} = {b}^{2} - 4 a c$
And:
$2 a x + b = \pm \sqrt{{b}^{2} - 4 a c}$
And finally:
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
Poetry in algebra!