Completing the Square of a Binomial Expression

Solving Quadratic Equations By Completing the Square

So far, we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called ‘completing the square.’

Completing The Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation \({\left(y-7\right)}^{2}=12\) because the left side was a perfect square.

We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form \({\left(x-k\right)}^{2}\) in order to use the square root property.

What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?

Let’s study the binomial square pattern we have used many times. We will look at two examples.

Binomial Squares Pattern

If \(a,b\) are real numbers,

\({\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}\)

\({\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}\)

We can use this pattern to “make” a perfect square.

We will start with the expression \({x}^{2}+6x\). Since there is a plus sign between the two terms, we will use the \({\left(a+b\right)}^{2}\) pattern.

Notice that the first term of \({x}^{2}+6x\) is a square, \({x}^{2}\).

We now know \(a=x\).

What number can we add to \({x}^{2}+6x\) to make a perfect square trinomial?

The middle term of the Binomial Squares Pattern, \(2ab\), is twice the product of the two terms of the binomial. This means twice the product of \(x\) and some number is \(6x\). So, two times some number must be six. The number we need is \(\frac{1}{2}·6=3.\) The second term in the binomial, \(b,\) must be 3.

We now know \(b=3\).

Now, we just square the second term of the binomial to get the last term of the perfect square trinomial, so we square three to get the last term, nine.

We can now factor to

So, we found that adding nine to \({x}^{2}+6x\) ‘completes the square,’ and we write it as \({\left(x+3\right)}^{2}\).

Example

Complete the square to make a perfect square trinomial. Then, write the result as a binomial square.

\({x}^{2}+14x\)

Solution

The coefficient of x is 14.
\(\begin{array}{}\\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {\left(\frac{1}{2}\cdot 14\right)}^{2}\hfill \\ \hfill {\left(7\right)}^{2}\hfill \\ \hfill 49\hfill \end{array}\)
Add 49 to the binomial to complete the square.	\({x}^{2}+14x+49\)
Rewrite as a binomial square.	\({\left(x+7\right)}^{2}\)

Example

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. \({m}^{2}-26m\)

Solution

The coefficient of m is −26.
\(\begin{array}{}\\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {\left(\frac{1}{2}\cdot \left(\text{−}26\right)\right)}^{2}\hfill \\ \hfill {\left(\text{−}13\right)}^{2}\hfill \\ \hfill 169\hfill \end{array}\)
Add 169 to the binomial to complete the square.	\({m}^{2}-26m+169\)
Rewrite as a binomial square.	\({\left(m-13\right)}^{2}\)

Example

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

\({u}^{2}-9u\)

Solution

The coefficient of u is −9.
\(\begin{array}{}\\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {\left(\frac{1}{2}\cdot \left(\text{−}9\right)\right)}^{2}\hfill \\ \hfill {\left(-\frac{9}{2}\right)}^{2}\hfill \\ \hfill \frac{81}{4}\hfill \end{array}\)
Add \(\frac{81}{4}\) to the binomial to complete the square.	\({u}^{2}-9u+\frac{81}{4}\)
Rewrite as a binomial square.	\({\left(u-\frac{9}{2}\right)}^{2}\)

Example

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

\({p}^{2}+\frac{1}{2}p\)

Solution

The coefficient of p is \(\frac{1}{2}\).
\(\begin{array}{}\\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {\left(\frac{1}{2}\cdot \frac{1}{2}\right)}^{2}\hfill \\ \hfill {\left(\frac{1}{4}\right)}^{2}\hfill \\ \hfill \frac{1}{16}\hfill \end{array}\)
Add \(\frac{1}{16}\) to the binomial to complete the square.	\({p}^{2}+\frac{1}{2}p+\frac{1}{16}\)
Rewrite as a binomial square.	\({\left(p+\frac{1}{4}\right)}^{2}\)