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By the end of this section, you will be able to:
  • Complete the square of a binomial expression
  • Solve quadratic equations of the form x 2 + b x + c = 0 by completing the square
  • Solve quadratic equations of the form a x 2 + b x + c = 0 by completing the square

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

  1. Simplify ( x + 12 ) 2 .
    If you missed this problem, review [link] .
  2. Factor y 2 18 y + 81 .
    If you missed this problem, review [link] .
  3. Factor 5 n 2 + 40 n + 80 .
    If you missed this problem, review [link] .

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.’

Complete the square of a binomial expression

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

( y 7 ) 2 = 12 y 7 = ± 12 y 7 = ± 2 3 y = 7 ± 2 3

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

x 2 10 x + 25 = 18 ( x 5 ) 2 = 18

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.

( x + 9 ) 2 ( x + 9 ) ( x + 9 ) x 2 + 9 x + 9 x + 81 x 2 + 18 x + 81 ( y 7 ) 2 ( y 7 ) ( y 7 ) y 2 7 y 7 y + 49 y 2 14 y + 49

Binomial squares pattern

If a , b are real numbers,

( a + b ) 2 = a 2 + 2 a b + b 2
( a b ) 2 = a 2 2 a b + b 2

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

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

a 2 + 2 a b + b 2 = ( a + b ) 2

Notice that the first term of x 2 + 6 x is a square, x 2 .

We now know a = x .

What number can we add to x 2 + 6 x to make a perfect square trinomial?

The image shows the expression a squared plus two a b plus b squared. Below it is the expression x squared plus six x plus a blank space. The x squared is below the a squared, the six x is below two a b and the blank is below the b squared.

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

The image is similar to the image above. It shows the expression a squared plus two a b plus b squared. Below it is the expression x squared plus two times three times x plus a blank space. The x squared is below the a squared, the two times three times x is below two a b and the blank is below the b squared.

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.

The image shows the expression a squared plus two a b plus b squared. Below it is the expression x squared plus six x plus nine.

We can now factor to

The image shows the expression quantity a plus b squared. Below it is the expression quantity x plus three squared.

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

Complete a square.

To complete the square of x 2 + b x :

  1. Identify b , the coefficient of x .
  2. Find ( 1 2 b ) 2 , the number to complete the square.
  3. Add the ( 1 2 b ) 2 to x 2 + b x .

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

x 2 + 14 x

Solution

The coefficient of x is 14. .
Find ( 1 2 b ) 2 . ( 1 2 14 ) 2 ( 7 ) 2 49
Add 49 to the binomial to complete the square. x 2 + 14 x + 49
Rewrite as a binomial square. ( x + 7 ) 2
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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

y 2 + 12 y

( y + 6 ) 2

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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

z 2 + 8 z

( z + 4 ) 2

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Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. m 2 26 m

Solution

The coefficient of m is −26. The image shows the expression m squared minus 26 m with x squared plus b x written above it. The coefficient of m is negative 26 so b is negative 26. Find half of b and square it. Half of negative 26 is negative 13 and negative 13 squared is 169. Add 169 to the binomial to complete the square and get the expression m squared minus 26 m plus 169 which is the quantity m minus 13 squared.
Find ( 1 2 b ) 2 . ( 1 2 ( 26 ) ) 2 ( 13 ) 2 169
Add 169 to the binomial to complete the square. m 2 26 m + 169
Rewrite as a binomial square. ( m 13 ) 2
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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