Completing the Square

Question 1

Problem 1) 2x^2 + 142 = -36x

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Question 2

Step 1?

Answer 2

Divide by Coefficient of x^2

2x^2 + 142 = -36x
————— = ——–
2 2

x^2 + 71 = -18x

Question 3

Step 2?

Answer 3

Move “x” Term to the Left

x^2 + 71 = -18x
+ 18x = + 18x

x^2 + 18x + 71 = 0

Question 4

Step 3?

Answer 4

Move Constant Term (“c”) to the Right

x^2 + 18x + 71 = 0
- 71 = - 71

x^2 + 18x = -71

Question 5

Step 4?

Answer 5

Find the Constant to Add: (b/2)^2

(b/2)^2 ➜ (18/2)^2 ➜9^2 ➜ 81

Question 6

Step 5?

Answer 6

Add the Constant (81 in this case) to Both Sides

x^2 + 18x + 81 = -71 +81
x^2 + 18x + 81 = 10 ➜ Combine Constants
(x + 9) (x + 9) = 10 ➜ Factor Trinomial
(x + 9)^2 = 10 ➜ Write as Square Binomial

Question 7

Step 6?

Answer 7

Square Root Both Sides:

√(x + 9)^2 = √10
x + 9 = ± √10

Question 8

Step 7?

Answer 8

Solve for X:

x + 9 = ± √10
- 9 = - 9
x = -9 ± √10

Question 9

Final Answer?

Answer 9

x = -9 ± √10

Question 10

Problem 2) x^2 - 12x + 21 = 0

Question 11

Step 1?

Answer 11

Move Constant Term to the Right

x^2 - 12x + 21 = 0
-21 = -21

x^2 - 12x = -21

Question 12

Step 2?

Answer 12

Find the Constant to add: (b/2)^2

(b/2)^2 ➜ (-12/2)^2 ➜ (-6)^2 ➜ 36

Question 13

Step 3?

Answer 13

Add Constant (Which is 36 in this case) to BOTH Sides

x^2 - 12x + 36 = -21 + 36
x^2 - 12x + 36 = 15 ➜ Combine Constants
(x - 6) (x - 6) = 15 ➜ Factor Trinomial
(x -6)^2 = 15 ➜ Write as Square Binomial

Question 14

Step 4?

Answer 14

Square Root Both Sides

√(x - 6)^2 = √15
x - 6 = ± √15

Question 15

Step 5?

Answer 15

Solve for X

x - 6 = ± √15
+ 6 = + 6

x = 6 ± √15

Question 16

Final Answer?

Answer 16

6 ± √15