1. 9.4 Solve by Completing the Square

2. What We Will Learn
Complete the square
Solve by completing the square

3. Needed Vocab
Completing the square: adding a constant to an expression to turn into a perfect square trinomial

4. Ex. 1 Completing the Square
Complete the square:
A. 𝑥 2 +6𝑥
𝑏 2 = 6 2
b = 3
3 2
Square = 9
𝑥 2 +6𝑥+9
Steps
1. find the b; 𝑏 2
2. square the b over 2; 𝑏 2 2
3. add the square to the end of the expression
B. 𝑥 2 −9𝑥
−9 2
− = 81 4
𝑥 2 −9𝑥+ 81 4

5. Your Practice Complete the square: 𝑥 2 +10𝑥 𝑏= 10 2 𝑏=5 𝑠𝑞𝑢𝑎𝑟𝑒= 5 2
𝑏= 10 2
𝑏=5
𝑠𝑞𝑢𝑎𝑟𝑒=
𝑠𝑞𝑢𝑎𝑟𝑒=25
𝑥 2 +10𝑥+25
Complete the square:
𝑥 2 −7𝑥
𝑏= −7 2
𝑠𝑞𝑢𝑎𝑟𝑒= −
𝑠𝑞𝑢𝑎𝑟𝑒= 49 2
𝑥 2 −7𝑥+ 49 2

6. Ex. 2 Solving 𝑥 2 +𝑏𝑥=𝑑 Steps 1. TAKE OUT GCF!! 2. complete the square
3. add square to both sides
4. combine like terms
5. factor
Square root front and back number, use sign of middle 𝑥∓
If fraction use square root of top and bottom
6. square root both sides
Remember there is a positive and negative answer
7. solve for letter

7. Ex. 2 Cont. 𝑥−8 2 =49 𝑥−8 2 = 49 𝑥−8=∓7 𝑥−8=7 𝑥−8=−7 +8 +8 +8 +8
Use completing the square
Solve 𝑥 2 −16𝑥=−15
𝑏= −16 2
𝑏=−8
𝑠𝑞𝑢𝑎𝑟𝑒= −8 2
𝑠𝑞𝑢𝑎𝑟𝑒=64
𝑥 2 −16𝑥+64=−15+64
𝑥 2 −16𝑥+64=49
𝑥−8 2 =49
𝑥−8 2 = 49
𝑥−8=∓7
𝑥−8= 𝑥−8=−7
𝑥= 𝑥=1

8. Ex. 2 Cont. Solve 𝑥 2 −2𝑥=3 𝑏= −2 2 𝑏=−1 𝑠𝑞𝑢𝑎𝑟𝑒= −1 2 𝑠𝑞𝑢𝑎𝑟𝑒=1
𝑏= −2 2
𝑏=−1
𝑠𝑞𝑢𝑎𝑟𝑒= −1 2
𝑠𝑞𝑢𝑎𝑟𝑒=1
𝑥 2 −2𝑥+1=3+1
𝑥 2 −2𝑥+1=4
𝑥−1 2 =4
𝑥−1 2 = 4
𝑥−1=∓2
𝑥−1= 𝑥−1=−2
𝑥= 𝑥=−1

9. Your Practice Solve 𝑥 2 −4𝑥=−2 𝑏= −4 2 𝑏=−2 𝑠𝑞𝑢𝑎𝑟𝑒= −2 2 𝑠𝑞𝑢𝑎𝑟𝑒=4
𝑏= −4 2
𝑏=−2
𝑠𝑞𝑢𝑎𝑟𝑒= −2 2
𝑠𝑞𝑢𝑎𝑟𝑒=4
𝑥 2 −4𝑥+4=−2+4
𝑥 2 −4𝑥+4=2
𝑥−2 2 =2
𝑥−2 2 = 2
Going to be a decimal
𝑥−2=∓ 2
𝑥−2= 𝑥−2=− 2
𝑥= 𝑥=.59

10. Ex. 3 Solving 𝑎𝑥 2 +𝑏𝑥+𝑐=0 Solve 2𝑥 2 +20𝑥−8=0 +8 +8 2𝑥 2 +20𝑥=8
Take a 2 out; remember that means divide EVERYTHING by 2
𝑥 2 +10𝑥=4
𝑥 2 +10𝑥+25=4+25
𝑥 2 +10𝑥+25=29
𝑥+5 2 =29
Steps
1. move number with no letter over = sign
2. same steps as example 2
𝑥 = 29
𝑥+5=∓ 29
𝑥+5= 𝑥+5=− 29
− − − −5
𝑥= 𝑥=−10.39

11. Your Practice 𝑔−4 2 = 25 𝑔−4=∓5 𝑔−4=5 𝑔−4=−5 +4 +4 +4 +4 𝑔=9 𝑔=−1
Solve 3𝑔 2 −24𝑔−27=0
3𝑔 2 −24𝑔=27
𝑔 2 −8𝑔=9
𝑔 2 −8𝑔+16=9+16
𝑔 2 −8𝑔+16=25
𝑔−4 2 =25
𝑔−4 2 = 25
𝑔−4=∓5
𝑔−4= 𝑔−4=−5
𝑔= 𝑔=−1