Solve each equation by completing the square

Question

 x^2 + 14x + 40 = 0

A) {-6, -10} B) {9, -3} C) {8, -4} D) {-4, -10}

 

7) n^2 - 16n + 63 = 0

A) {9, 7} B) {-1, -7} C) {10, 2} D) {-5 + √69, -5 - √69}

 

6) n^2 - 10n + 21 = 0

A) {5 + √46, 5 - √46} B) {2, -6} C) {7, 3} D) {-7 + 6 √3, -7 - 6 √3}

Answer 1

1)

The equation x^2 + 14x + 40 = 0

Separate variables and constants aside by subtracting 1 to each side.

x^2 + 14x  =  - 40

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

x^2 + 14x + 7^2  =  - 40 + 7^2

(x + 7)^2  =  - 40 + 49

(x + 7)^2  =  9

Take square root on both sides.

(x + 7)   =   ± √9

(x + 7)   =   ± 3

x   =  - 7 ± 3

x   =  - 7 + 3    ;      x   =  - 7 - 3

x   =  - 4          ;      x   =  - 10

Solutions are x = - 4 and x = - 10

 

2)

The equation  n^2 - 16n + 63 = 0

Separate variables and constants aside by subtracting 1 to each side.

n^2 - 16n  =  - 63

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

n^2 - 16n + 8^2  =  - 63 + 8^2

(n - 8)^2  =  - 63 + 64

(n - 8)  =  1

Take square root on both sides.

(n - 8)  =  ± √1

(n - 8)  =  ± 1

n  =  8 ± 1

n  =  8 + 1      ;      n  =  8 - 1

n  =  9      ;      n  =  7

Solutions are n  =  9  and  n  =  7

 

3)

The equation  n^2 - 10n + 21 = 0

Separate variables and constants aside by subtracting 1 to each side.

n^2 - 10n  =  - 21

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

n^2 - 10n + 5^2  =  - 21 + 5^2

(n - 5)^2  =  - 21 + 25

(n - 5)  =  4

Take square root on both sides.

(n - 5)  =  ± √4

(n - 5)  =  ± 2

n  =  5 ± 2

n  =  5 + 2      ;      n  =  5 - 2

n  =  7            ;      n  =  3

Solutions are n  =  7  and  n  =  3

 

Answer :

1) Option D

2) Option A

3) Option C