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. This article or section needs to be written in plain English, using plain English that most of our readers can understand. Articles that depend excessively on technical terms accessible only to specialists are useless for our purposes, so writers are admonished to avoid jargon