n^2+13n+27=-5

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Solution for n^2+13n+27=-5 equation:


Simplifying
n2 + 13n + 27 = -5

Reorder the terms:
27 + 13n + n2 = -5

Solving
27 + 13n + n2 = -5

Solving for variable 'n'.

Reorder the terms:
27 + 5 + 13n + n2 = -5 + 5

Combine like terms: 27 + 5 = 32
32 + 13n + n2 = -5 + 5

Combine like terms: -5 + 5 = 0
32 + 13n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-32' to each side of the equation.
32 + 13n + -32 + n2 = 0 + -32

Reorder the terms:
32 + -32 + 13n + n2 = 0 + -32

Combine like terms: 32 + -32 = 0
0 + 13n + n2 = 0 + -32
13n + n2 = 0 + -32

Combine like terms: 0 + -32 = -32
13n + n2 = -32

The n term is 13n.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13n + 42.25 + n2 = -32 + 42.25

Reorder the terms:
42.25 + 13n + n2 = -32 + 42.25

Combine like terms: -32 + 42.25 = 10.25
42.25 + 13n + n2 = 10.25

Factor a perfect square on the left side:
(n + 6.5)(n + 6.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(n + 6.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

n + 6.5 = 3.201562119 Simplifying n + 6.5 = 3.201562119 Reorder the terms: 6.5 + n = 3.201562119 Solving 6.5 + n = 3.201562119 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = 3.201562119 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = 3.201562119 + -6.5 n = 3.201562119 + -6.5 Combine like terms: 3.201562119 + -6.5 = -3.298437881 n = -3.298437881 Simplifying n = -3.298437881

Subproblem 2

n + 6.5 = -3.201562119 Simplifying n + 6.5 = -3.201562119 Reorder the terms: 6.5 + n = -3.201562119 Solving 6.5 + n = -3.201562119 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = -3.201562119 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = -3.201562119 + -6.5 n = -3.201562119 + -6.5 Combine like terms: -3.201562119 + -6.5 = -9.701562119 n = -9.701562119 Simplifying n = -9.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-3.298437881, -9.701562119}

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