Help me solve these equations by completing the square please?

Question

Round to the nearest tenth if necessary. Detailed explanation to give complete understanding would be preferred.

 

9. x^2 + 20x +11 = -8

 

10. x^2 - 1 +5x

 

11. a^2 = 22a + 23

 

12. m^2 - 8m =-7

Answer 1

When we complete the square, we want to have a binomial squared only on one side of the equation.

 

Question 9

 

x² + 20x + 11 = -8

 

Let's find a binomial on the left side so when we square it and expand it, the first term is x² and middle term is 20x. 

 

(x + 10)² = (x + 10)(x + 10) = x² + 20x + 100

 

 

Add 89 on both sides of the equation. We obtain

 

x² + 20x + 11 + 89 = -8 + 89

x² + 20x + 100 = 81

(x + 10)² = 81

 

 

Square-root both sides of equation to undo the exponent on the left side of equation.

 

x + 10 = ±√(81)

x = -10 ± √(81)

x = -10 ± 9

 

x = -10 - 9        and       x = -10 + 9

 

x = -19             and       x = -1

 

 

Question 11

 

Subtract 22a on both sides of equation.

 

a² - 22a = 23

 

We want the complete square of (a - 11)², since this is equal to a² - 22a + 121 when expanded.

 

Add 121 on both sides of equation.

 

a² - 22a + 121 = 23 + 121

(a - 11)² = 144

a - 11 = ±√(144)

a = 11 ± √(144)

a = 11 ± 12

 

a = 11 - 12             and           a = 11 + 12

a = -1                    and           a = 23

 

 

Try the last problem on your own.