solve each equation by completing the square.

Question

n^2+12n+37=2 

k^2+8k-47=7

Answer 1

The given equation n²+12n+37 = 2

Add 36 to each side

n²+12n+37+36 = 36+2

n²+12n+36 = 38 - 37

n²+12n+36 = 1

Now, n²+12n+36 is the form of  ( a+b)²= a²+b²+2ab

(n+6)²= 1

Now  taking square root  on both side

n+6 = ±1

Therefore n = - 5 and n = -7

So the factors of the given quadratic equation is ( n+5) and ( n+7).

The given equation is k²+8k - 47 = 7

Add 16 to each side

k²+8k - 47+16 = 7+16

k²+8k +16 - 47 = 23

k²+8k +16 = 70

Now, k²+8k +16  is the form of  ( a+b)²= a²+b²+2ab

( k+4)²= 70

k +4 = ±√70

Therefore k = ( - 4+√70) ,( -4 - √70)

So the factors of the given quadratic equation is( k+4+√70) and .( k +4-√70).

Answer 2

• The equation is n² + 12n + 37 = 2.

Separate variables and constants aside by subtracting 37 to each side.

n² + 12n = 2 - 37

n² + 12n = - 35

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

 Here n coefficient = 12. so, (half the n coefficient)² = (12/2)²= 36

Add 36 to each side.

n² + 12n + 36 = - 35 + 36

(n + 6)² = 1

n + 6 = ± 1

 n = ± 1 - 6

 n = 1 - 6 and n = - 1 - 6

⇒ n = - 5 and n = - 7.

The solutions are n = - 5 and n = - 7.

 

• The equation is k² + 8k - 47 = 7.

Separate variables and constants aside by adding 47 to each side.

k² + 8k = 7 + 47

k² + 8k = 54

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

 Here k coefficient = 8. so, (half the k coefficient)² = (8/2)²= 16

Add 16 to each side.

k² + 8k + 16 = 54 + 16

(k + 4)² = 70

k + 4 = ± √70

 k = ± √70 - 4

 ⇒ k = √70 - 4 and k = - √70 - 4.

The solutions are k = √70 - 4 and k = - √70 - 4.