Solve by completing the square

Solve by completing the square could take a little bit more time to do than solving by factoring. Keep reading to see a step-by-step explanation.

Before showing more examples, you need to understand what a perfect square trinomial is.

Binomial × same Binomial = Perfect square trinomial

(x + 4) × (x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16

(x + a) × (x + a) = x² + ax + ax + a² = x² + 2ax + a²

Important observation:

What is the relationship between the coefficient of the second term and the last term?

For x² + 8x + 16, the coefficient of the second term is 8 and the last term is 16

(8/2)² = 4² = 16

For, x² + 2ax + a², the coefficient of the second term is 2a and the last term is a²

(2a/2)² = a²

So, what is the relationship?

The last term is obtained by dividing the coefficient of the second term by 2 and squaring the result.

Now, let's say you have x² + 20x and you want to find the last term to make the whole thing a perfect square trinomial.

Just do (20/2)² = 10². The perfect square trinomial is x² + 20x + 10²= (x + 10) × (x + 10)

Examples showing how to solve by completing the square.

The steps to follow are straightforward as you can see in the example shown below. 

More examples showing how to solve by completing the square

Example #1:

Solve by completing the square x² + 6x + 8 = 0

x² + 6x + 8 = 0

Subtract 8 from both sides of the equation.

x² + 6x + 8 - 8 = 0 - 8

x² + 6x = - 8

To complete the square, always do the following 24 hours a day 365 days a year. It is never going to change when you solve by completing the square!

You are basically looking for a term to add to x² + 6x that will make it a perfect square trinomial.

To this end, get the coefficient of the second term, divide it by 2 and raise it to the second power.

The second term is 6x and the coefficient is 6.

6/2 = 3 and after squaring 3, we get 3²

x² + 6x = - 8

Add 3² to both sides of the equation above

x² + 6x + 3² = - 8 + 3²

(x + 3)² = -8 + 9

(x + 3)² = 1

Take the square root of both sides

√((x + 3)²) = √(1)

x + 3 = ±1

When x + 3 = 1, x = -2

When x + 3 = -1, x = -4

Example #2:

Solve by completing the square x² + -6x + 8 = 0 instead of x² + 6x + 8 = 0

The second term this time is -6x and the coefficient is -6.

-6/2 = -3 and after squaring -3, we get (-3)² = 9

x² + -6x = - 8

Add (-3)² to both sides of the equation above

x² + -6x + (-3)² = - 8 + (-3)²

(x + -3)² = -8 + 9

(x + -3)² = 1

Take the square root of both sides

√((x + -3)²) = √(1)

x + -3 = ±1

When x + -3 = 1, x = 4

When x + -3 = -1, x = 2

Example #3:

Solve by completing the square 3x² + 8x + -3 = 0

3x² + 8x + -3 = 0

Divide everything by 3. Always do that when the coefficient of the first term is not 1

(3/3)x²+ (8/3)x + -3/3 = 0/3

x²+ (8/3)x + -1 = 0

Add 1 to both sides of the equation.

x² + (8/3)x + -1 + 1 = 0 + 1

x² + (8/3)x = 1

The second term is (8/3)x and the coefficient is 8/3.

8/3 ÷ 2 = 8/3 × 1/2 = 8/6 and after squaring 8/6, we get (8/6)²

x² + (8/3)x = 1

Add (8/6)² to both sides of the equation above

x² + (8/3)x + (8/6)² = 1 + (8/6)²

(x + 8/6)² = 1 + 64/36

(x + 8/6)² = 36/36 + 64/36 = (36 + 64)/36 = 100/36

Take the square root of both sides

√((x + 8/6)²) = √(100/36)

x + 8/6 = ±10/6

x + 8/6 = 10/6

x = 10/6 - 8/6 = 2/6 = 1/3

x + 8/6 = - 10/6

x = -10/6 - 8/6 = -18/6 = -3

To solve by completing the square can become quickly hard as shown in example #3