5.7 – Completing the Square

Completing the squareEasy case x² + 8x – 5 = 9

• x² + 8x – 5 = 9

• x² + 8x = 14

• x² + 8x + 16 = 14 + 16

• (x+4)² = 30

• (x+4) = ±√30

• x = -4 ±√30

162

8

2

b22

Add the numbers to right, variables to the left

Add 16 to both sides

Factor the left side

Square root both sides

Get x by itself

Completing the squareMedium case 2x² + 12x – 1 = 20

• 2x² + 12x – 1 = 20

• 2x² + 12x = 21

• x² + 6x = 10.5

• x² + 6x + 9 = 10.5 + 9

• (x+3)² = 19.5

• (x+3) = ±√19.5

• x = -3 ±√30

Add 92

6

2

b22

a must be =1. Divide everything by a if it is not already = 1

Hard case2x² + 9x = 16

• 2x² + 9x = 16• x² + 9/2 x + 81/16 =

8 + 81/16• (x + 9/4)²= 8 + 81/16• (x + 9/4)²= 128/16 + 81/16• (x + 9/4)²= 209/16• (x + 9/4) = ±√(209/16)• (x + 9/4) = ±√209/4• x = -9/4 ± √209/4

Add

16

81

4

9

2

2/9

2

b222

4

2099x

WHEW!

Try these on your own:

a)x2 -12x +5 =0

b) x2 +8x +36 =0

316x

524 ix