## Derivation of the abc Quadratic Formula

$ax^2 +bx+c=0$
can be solved for
$x$
using the formula
$x= \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
.
We can derive this formula by completing the square.
Starting from
$ax^2 +bx+c=0$
, multiply by
$a$
.
$a^2x^2 +abx+ac=0$

Ignore
$ac$
for the moment and complete the square for
$a^2x^2 +abx$
.
$a^2x^2 +abx+ac=((ax+\frac{b}{2})^2 - (\frac{b}{2})^2)+ac=0$

$(\frac{b}{2})^2$
and subtract
$ac$
.
$(ax+\frac{b}{2})^2 = (\frac{b}{2})^2-ac =\frac{b^2}{4}-ac= \frac{b^2-4ac}{4}$

Square root both sides.
$ax+\frac{b}{2} = \sqrt{(\frac{b^2-4ac}{4})}=\pm \frac{\sqrt{b^2-4ac}}{2}$

Now subtract
$\frac{b}{2}$
from both sides.
$ax =- \frac{b}{2} \sqrt{(\frac{b^2-4ac}{4})}= \frac{-b \pm\sqrt{b^2-4ac}}{2}=$

Finally divide both sides by
$a$
.
$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
.