n^2+n=152

Solution for n^2+n=152 equation:

Simplifying
n2 + n = 152

Reorder the terms:
n + n2 = 152

Solving
n + n2 = 152

Solving for variable 'n'.

Reorder the terms:
-152 + n + n2 = 152 + -152

Combine like terms: 152 + -152 = 0
-152 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '152' to each side of the equation.
-152 + n + 152 + n2 = 0 + 152

Reorder the terms:
-152 + 152 + n + n2 = 0 + 152

Combine like terms: -152 + 152 = 0
0 + n + n2 = 0 + 152
n + n2 = 0 + 152

Combine like terms: 0 + 152 = 152
n + n2 = 152

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 152 + 0.25

Reorder the terms:
0.25 + n + n2 = 152 + 0.25

Combine like terms: 152 + 0.25 = 152.25
0.25 + n + n2 = 152.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 152.25

Calculate the square root of the right side: 12.338962679

Break this problem into two subproblems by setting 
(n + 0.5) equal to 12.338962679 and -12.338962679.

Subproblem 1
n + 0.5 = 12.338962679

Simplifying
n + 0.5 = 12.338962679

Reorder the terms:
0.5 + n = 12.338962679

Solving
0.5 + n = 12.338962679

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 12.338962679 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 12.338962679 + -0.5
n = 12.338962679 + -0.5

Combine like terms: 12.338962679 + -0.5 = 11.838962679
n = 11.838962679

Simplifying
n = 11.838962679

Subproblem 2
n + 0.5 = -12.338962679

Simplifying
n + 0.5 = -12.338962679

Reorder the terms:
0.5 + n = -12.338962679

Solving
0.5 + n = -12.338962679

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -12.338962679 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -12.338962679 + -0.5
n = -12.338962679 + -0.5

Combine like terms: -12.338962679 + -0.5 = -12.838962679
n = -12.838962679

Simplifying
n = -12.838962679

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {11.838962679, -12.838962679}