Completing the square

Completing the square is a method for solving for the roots of the general quadratic equation:

$a x 2 + b x + c = 0$ , where $a \ne 0$

It is first taught using equations with "friendly" numbers in place of a, b, and c to get the student used to the process.

What one does is add and multiply by various carefully chosen constants to create an equation of the form:

d 2 x 2 + 2 d e x + e 2 = f

where d, e and f are constants expressed in a, b, and c.

This resolves to:

(d x + e) 2 = f

(grouping)

$dx + e = \pm \sqrt{f}$ (take square root)

$dx = -e \pm \sqrt{f}$ (subtract e)

$x = \frac{-e \pm \sqrt{f}}{d}$ (divide by e)

By then applying the process to the general equation, we can derive the quadratic formula:

a x 2 + b x + c = 0

(given)

4 a 2 x 2 + 4 a b x + 4 a c = 0

(multiply by 4a)

4 a 2 x 2 + 4 a b x = - 4 a c

(subtract 4ac)

4 a 2 x 2 + 4 a b x + b 2 = - 4 a c + b 2

(add b^2)

(2 a x + b) 2 = b 2 - 4 a c

(group each side)

$2ax + b = \pm \sqrt{b^2 - 4ac}$ (take sqaure root, allow for both roots)

$2ax = -b \pm \sqrt{b^2 - 4ac}$ (subtract b)

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (divide by 2a)

We can now determine the real or imaginary roots of any quadratic equation by simply inserting a, b, and c into the formula.