Completing the Square MM3G2
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
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
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 3 2 2 = 9 4 𝑥 2 +3𝑥+ 9 4 =6+ 9 4 𝑥 2 +3𝑥+ 9 4 = 33 4 𝑥+ 3 2 2 = 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
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= 11 2 +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
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