9.4 Solve by Completing the Square

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

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

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 −9 2 2 = 81 4 𝑥 2 −9𝑥+ 81 4

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

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∓ 2 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

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=7 𝑥−8=−7 +8 +8 +8 +8 𝑥=15 𝑥=1

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=2 𝑥−1=−2 +1 +1 +1 +1 𝑥=3 𝑥=−1

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= 2 𝑥−2=− 2 +2 +2 +2 +2 𝑥=3.41 𝑥=.59

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+5 2 = 29 𝑥+5=∓ 29 𝑥+5= 29 𝑥+5=− 29 −5 −5 −5 −5 𝑥=.39 𝑥=−10.39

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 +27 +27 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=5 𝑔−4=−5 +4 +4 +4 +4 𝑔=9 𝑔=−1