n^2+2n-8=78

Solution for n^2+2n-8=78 equation:

Simplifying
n2 + 2n + -8 = 78

Reorder the terms:
-8 + 2n + n2 = 78

Solving
-8 + 2n + n2 = 78

Solving for variable 'n'.

Reorder the terms:
-8 + -78 + 2n + n2 = 78 + -78

Combine like terms: -8 + -78 = -86
-86 + 2n + n2 = 78 + -78

Combine like terms: 78 + -78 = 0
-86 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '86' to each side of the equation.
-86 + 2n + 86 + n2 = 0 + 86

Reorder the terms:
-86 + 86 + 2n + n2 = 0 + 86

Combine like terms: -86 + 86 = 0
0 + 2n + n2 = 0 + 86
2n + n2 = 0 + 86

Combine like terms: 0 + 86 = 86
2n + n2 = 86

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 86 + 1

Reorder the terms:
1 + 2n + n2 = 86 + 1

Combine like terms: 86 + 1 = 87
1 + 2n + n2 = 87

Factor a perfect square on the left side:
(n + 1)(n + 1) = 87

Calculate the square root of the right side: 9.327379053

Break this problem into two subproblems by setting 
(n + 1) equal to 9.327379053 and -9.327379053.

Subproblem 1
n + 1 = 9.327379053

Simplifying
n + 1 = 9.327379053

Reorder the terms:
1 + n = 9.327379053

Solving
1 + n = 9.327379053

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + n = 9.327379053 + -1

Combine like terms: 1 + -1 = 0
0 + n = 9.327379053 + -1
n = 9.327379053 + -1

Combine like terms: 9.327379053 + -1 = 8.327379053
n = 8.327379053

Simplifying
n = 8.327379053

Subproblem 2
n + 1 = -9.327379053

Simplifying
n + 1 = -9.327379053

Reorder the terms:
1 + n = -9.327379053

Solving
1 + n = -9.327379053

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + n = -9.327379053 + -1

Combine like terms: 1 + -1 = 0
0 + n = -9.327379053 + -1
n = -9.327379053 + -1

Combine like terms: -9.327379053 + -1 = -10.327379053
n = -10.327379053

Simplifying
n = -10.327379053

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {8.327379053, -10.327379053}