### Completing the Square

You saw in the last section that if you have a quadratic equation of the form

Simplify to get:

So what do you do with an equation that isn’t written in this nice form? In this section, you’ll learn how to rewrite any quadratic equation in this form by **completing the square.**

**Complete the Square of a Quadratic Expression**

Completing the square lets you rewrite a quadratic expression so that it contains a perfect square trinomial that you can factor as the square of a binomial.

Remember that the square of a binomial takes one of the following forms:

So in order to have a perfect square trinomial, we need two terms that are perfect squares and one term that is twice the product of the square roots of the other terms.

#### Completing the Square

1. Complete the square for the quadratic expression

To complete the square we need a constant term that turns the expression into a perfect square trinomial. Since the middle term in a perfect square trinomial is always 2 times the product of the square roots of the other two terms, we re-write our expression as:

We see that the constant we are seeking must be

**Answer:** By adding 4 to both sides, this can be factored as:

Notice, though, that we just changed the value of the whole expression by adding 4 to it. If it had been an equation, we would have needed to add 4 to the other side as well to make up for this.

Also, this was a relatively easy example because

2. Complete the square for the quadratic expression

Factor the coefficient of the

Re-write the expression:

We complete the square by adding the constant

Factor the perfect square trinomial inside the parenthesis:

The expression **“completing the square”** comes from a geometric interpretation of this situation. Let’s revisit the quadratic expression in Example 1:

We can think of this expression as the sum of three areas. The first term represents the area of a square of side

We can combine these shapes as follows:

We obtain a square that is not quite complete. To complete the square, we need to add a smaller square of side length 2.

We end up with a square of side length **completing the square** with an example.

#### Solving for Unknown Values

Solve the following quadratic equation:

Divide all terms by the coefficient of the

Rewrite:

In order to have a perfect square trinomial on the right-hand-side we need to add the constant **both** sides of the equation:

Factor the perfect square trinomial and simplify:

Take the square root of both sides:

**Answer:**

**Solving Quadratic Equations in Standard Form**

If an equation is in standard form

Solve the following quadratic equation:

Move the constant to the other side of the equation:

Rewrite:

Add the constant

Factor the perfect square trinomial and simplify:

Take the square root of both sides:

**Answer:**

### Example

#### Example 1

Solve the following quadratic equation:

Divide all terms by the coefficient of the

Rewrite:

In order to have a perfect square trinomial on the right-hand-side we need to add the constant **both** sides of the equation:

Factor the perfect square trinomial and simplify:

Take the square root of both sides:

\begin{align*}x - 11 &= \sqrt{16} && \text{and} && x - 11 = - \sqrt{ 16}\\ x &= 11 + \sqrt{16} =15 && \text{and} && x =11 - \sqrt{4}= 7\end{align*}

**Answer:** \begin{align*}x = 15\end{align*} and \begin{align*}x =7\end{align*}

### Review

Complete the square for each expression.

- \begin{align*}x^2 + 5x\end{align*}
- \begin{align*}x^2 - 2x\end{align*}
- \begin{align*}x^2 + 3x\end{align*}
- \begin{align*}x^2 - 4x\end{align*}
- \begin{align*}3x^2 + 18x\end{align*}
- \begin{align*}2x^2 - 22x\end{align*}
- \begin{align*}8x^2 - 10x\end{align*}
- \begin{align*}5x^2 + 12x\end{align*}

Solve each quadratic equation by completing the square.

- \begin{align*}x^2 - 4x = 5\end{align*}
- \begin{align*}x^2 - 5x = 10\end{align*}
- \begin{align*}x^2 + 10x + 15 = 0\end{align*}
- \begin{align*}x^2 + 15x + 20 = 0\end{align*}
- \begin{align*}2x^2 - 18x = 0\end{align*}
- \begin{align*}4x^2 + 5x = -1\end{align*}
- \begin{align*}10x^2 - 30x - 8 = 0\end{align*}
- \begin{align*}5x^2 + 15x - 40 = 0\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 10.6.