Completing the Square

In algebra, completing the square is a mathematical method demonstrated by a man named al-Khwarzimi (780–850), who published a book that gave algebra its very name.

Suppose you were given the following mathematical expression: x2 + 16x. You have been asked to complete the square of this expression. How do you do that?

Here's how it's done:

1. Take the original expression...

$$x^2 + 16x$$

2. Factor out the x...

$$(x + 8)^2$$

3. Square the constant term (the term with no variable)...

$$8^2 = 64$$

(or $$8 * 8 = 64$$)

4. And apply it to the original expression. All squared terms remain positive when completing the square of an expression.

$$x^2 + 16x + 64$$

Congratulations! You've completed the square!

When There's a Coefficient
When coefficients are involved, the only major difference is the inclusion of one step: dividing the entire expression by that coefficient.

For instance, let's say you were given the following expression: 4x2 - 16x.

It's very easy, you know. All you have to do is divide the entire expression by 4.

$$4x^2 - 16x$$ becomes $$x^2 - 4x$$.

Then, it's all downhill from here. Complete the square by following the steps above, and your answer should come up as this:

$$x^2 - 4x + 4$$

When the Linear Term Is An Odd Number
Suppose you were given the expression x2 + 7x. Here's what you do when the linear term is an odd number:

1. Factor out the x and put 7 over 2...

From $$x^2 + 7x$$ to $$(x + \frac{7}{2}\,)^2$$...

2. Square the fraction (there's an easy way)...

$$(\frac{7}{2}\,)^2 = \frac{7 * 7}{2 * 2}\, = \frac{49}{4}\,$$

3. And apply it to the expression. If you want to, you can simplify the improper fraction into a mixed number or decimal.

$$x^2 + 7x + \frac{49}{4}\,$$, which is also $$x^2 + 7x + 12\frac{1}{4}\,$$ or $$x^2 + 7x + 12.25$$

There you have it!

Completing the Square to Solve Quadratic Equations
Al-Khwarzimi showed how to solve quadratic equations by finding the square by:

1. Rewriting the original problem...

$$x^2 - 6x - 5 = 0$$

2. Moving the variables to different sides...

$$-5 = -x^2 + 6x$$

3. Factoring the quadratic and linear terms to complete the square...

$$5 + 9 = x^2 - 6x + 9$$

$$14 = (x - 3)^2$$

4. Taking the square roots of both sides...

$$\pm \sqrt{14}\, = x - 3$$

5. And, finally, simplifying the whole thing.

$$3 \pm \sqrt{14}\, = x$$

This is a little trickier, though it's easy once you get used to it.

Now, Try This!
Complete the square to solve for x. Leave the answer in simplest radical form.

$$4x^2 + 4 = 32x + 8 $$