Completing the square

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

<math>ax^2 + bx + c = 0</math>, where <math>a \ne 0</math>

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:

<math>d^2x^2 + 2dex + e^2 = f</math> where d, e and f are constants expressed in a, b, and c.

This resolves to:

<math>(dx+e)^2 = f</math> (grouping)

<math>dx + e = \pm \sqrt{f}</math> (take square root)

<math>dx = -e \pm \sqrt{f}</math> (subtract e)

<math>x = \frac{-e \pm \sqrt{f}}{d}</math> (divide by e)

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

<math>ax^2 + bx + c = 0</math> (given)

<math>4a^2x^2 + 4abx + 4ac = 0</math> (multiply by 4a)

<math>4a^2x^2 + 4abx = -4ac</math> (subtract 4ac)

<math>4a^2x^2 + 4abx + b^2 = -4ac + b^2</math> (add b^2)

<math>(2ax + b)^2 = b^2 - 4ac</math> (group each side)

<math>2ax + b = \pm \sqrt{b^2 - 4ac}</math> (take square root, allow for both roots)

<math>2ax = -b \pm \sqrt{b^2 - 4ac}</math> (subtract b)

<math>x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}</math> (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.