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