1. Completing the Square
MM3G2

2. Steps for completing the square
Make sure the equation is in the proper form
Move the constant to the other side
Identify the coefficient of the x term
Half it, square it, add it to both sides
Factor the equation on the left
𝑥 2 +10𝑥+8=0
𝑥 2 +10𝑥=−8 10
10 2 =5
5 2 =25
𝑥 2 +10𝑥+25=−8+25
𝑥 2 +10𝑥+25=17
𝑥+5 2 =17

3. Example 2:
3 𝑥 2 +24𝑥+12=0
3 𝑥 2 +24𝑥+12=0 3
𝑥 2 +8𝑥+4=0
𝑥 2 +8𝑥=−4 8
8 2 =4
4 2 =16
𝑥 2 +8𝑥+16=−4+16
𝑥 2 +8𝑥+16=12
𝑥+4 2 =12
Make sure the equation is in the proper form
leading coefficient must be 1
Divide everything by the leading coefficient
Move the constant to the other side
Identify the coefficient of the x term
Half it, square it, add it to both sides
Factor the equation on the left

4. Example 3: 2 𝑥 2 +6𝑥−12=0 2 𝑥 2 +6𝑥−12=0 2 𝑥 2 +3𝑥−6=0 𝑥 2 +3𝑥=6 3 3 2
2 𝑥 2 +6𝑥−12=0 2
𝑥 2 +3𝑥−6=0
𝑥 2 +3𝑥=6 3
3 2
= 9 4
𝑥 2 +3𝑥+ 9 4 =6+ 9 4
𝑥 2 +3𝑥+ 9 4 = 33 4
𝑥 = 33 4
Make sure the equation is in the proper form
leading coefficient must be 1
Divide everything by the leading coefficient
Move the constant to the other side
Identify the coefficient of the x term
Half it, square it, add it to both sides
Factor the equation on the left

5. Example 4: 2 𝑥 2 −12𝑥−11=0 2 𝑥 2 −12𝑥−11=0 2 𝑥 2 −6𝑥− 11 2 =0
2 𝑥 2 −12𝑥−11=0 2
𝑥 2 −6𝑥− 11 2 =0
𝑥 2 −6𝑥= 11 2 -6
−6 2
−3 2 =9
𝑥 2 −6𝑥+9=
𝑥 2 −6𝑥+9= 29 2
𝑥−3 2 = 29 2
Make sure the equation is in the proper form
leading coefficient must be 1
Divide everything by the leading coefficient
Move the constant to the other side
Identify the coefficient of the x term
Half it, square it, add it to both sides
Factor the equation on the left

6. Example 5: 5 𝑥 2 =30𝑥−40 5 𝑥 2 −30𝑥=−40 5 𝑥 2 −30𝑥=−40 5 𝑥 2 −6𝑥=−8 -6
5 𝑥 2 −30𝑥=−40 5
𝑥 2 −6𝑥=−8 -6
−6 2
−3 2 =9
𝑥 2 −6𝑥+9=−8+9
𝑥 2 −6𝑥+9=1
𝑥−3 2 =1
Make sure the equation is in the proper form
leading coefficient must be 1
Divide everything by the leading coefficient
Move the constant to the other side
Identify the coefficient of the x term
Half it, square it, add it to both sides
Factor the equation on the left