
Solve by completing the square
Solve by completing the square could take a little bit more time to do than solving by factoring. However, the steps are straightforward.
Before showing examples, you need to understand what a perfect square trinomial is
Binomial × same Binomial = Perfect square trinomial
(x + 4) × (x + 4) = x^{2} + 4x + 4x + 16 = x^{2} + 8x + 16
(x + a) × (x + a) = x^{2} + ax + ax + a^{2} = x^{2} + 2ax + a^{2}
Important observation:
What is the relationship between the coefficient of the second term and the last term?
For x^{2} + 8x + 16, coefficient of the second term is 8 and the last term is 16
(8/2)^{2} = 4^{2} = 16
For, x^{2} + 2ax + a^{2}, the coefficient of the second term is 2a and the last term is a^{2}
(2a/2)^{2} = a^{2}
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 x^{2} + 20x and you want to find the last term to make the whole thing a perfect square trinomial
Just do (20/2)^{2} = 10^{2}. The perfect square trinomial is x^{2} + 20x + 10^{2}= (x + 10) × (x + 10)
Example #1:
Solve by completing the square x^{2} + 6x + 8 = 0
x^{2} + 6x + 8 = 0
Subtract 8 from both sides of the equation.
x^{2} + 6x + 8  8 = 0  8
x^{2} + 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 x^{2} + 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 3^{2}
x^{2} + 6x =  8
Add 3^{2} to both sides of the equation above
x^{2} + 6x + 3^{2} =  8 + 3^{2}
x^{2} + 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 = 2
When x + 3 = 1, x = 4
Example #2:
Solve by completing the square x^{2} + 6x + 8 = 0 instead of x^{2} + 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
x^{2} + 6x =  8
Add (3)^{2} to both sides of the equation above
x^{2} + 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 3x^{2} + 8x + 3 = 0
3x^{2} + 8x + 3 = 0
Divide everything by 3. Always do that when the coefficient of the first term is not 1
(3/3)x^{2}+ (8/3)x + 3/3 = 0/3
x^{2}+ (8/3)x + 1 = 0
Add 1 to both sides of the equation.
x^{2} + (8/3)x + 1 + 1 = 0 + 1
x^{2} + (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}
x^{2} + (8/3)x = 1
Add (8/6)^{2} to both sides of the equation above
x^{2} + (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 shwon in example #3

