Statement
Given the equation:
$$ ax^2 + bx + c = 0 $$
The values of $ x $ can be found using this formula:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Proof
Take the standard quadratic equation.
$$ ax^2 + bx + c = 0 $$
Divide every term by $ a $.
$$ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $$
Complete the square by adding and subtracting $ \left(\frac{b}{2a}\right)^2 $.
$$ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0 $$
$$ \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0 $$
Isolate the square.
$$ \left(x + \frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a} $$
Expand the brackets on the right hand side and combine the fractions.
$$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{4ac}{4a^2} $$
$$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $$
Take the square root on both sides. Note that this will result in a $ \pm $-sign on the right hand side.
$$ x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} $$
Now isolate the $ x $-term.
$$ x = - \frac{b}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} $$
Finally, simplify the result.
$$ x = \frac{-b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{\sqrt{4a^2}} $$
$$ x = \frac{-b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$