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 how to solve a quadratic equation by completing the square, you need to understand what a perfect square trinomial is and how to complete a square.

Perfect square trinomial = Binomial × same Binomial 

Examples

(x + 4) × (x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16 and x² + 8x + 16 is a perfect square trinomial

(x + a) × (x + a) = x² + ax + ax + a² = x² + 2ax + a² and x² + 2ax + a² is a perfect square trinomial.

How to complete a square by making an 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. 

Example #1:

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

Step-by-Step Explanations

x² + 6x + 8 = 0

Subtract 8 from both sides of the equation so that you can isolate the two terms containing variables.

x² + 6x + 8 - 8 = 0 - 8

x² + 6x = - 8

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 so you end up with the linear equations.

√((x + 3)2) = √(1)

x + 3 = ±1

When x + 3 = 1, x = -2

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

More examples showing how to solve by completing the square

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