n^2+n=5

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Solution for n^2+n=5 equation:


Simplifying
n2 + n = 5

Reorder the terms:
n + n2 = 5

Solving
n + n2 = 5

Solving for variable 'n'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + n + n2 = 0 + 5

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

Combine like terms: 0 + 5 = 5
n + n2 = 5

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 = 5 + 0.25

Reorder the terms:
0.25 + n + n2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + n + n2 = 5.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(n + 0.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
n + 0.5 = 2.291287847

Simplifying
n + 0.5 = 2.291287847

Reorder the terms:
0.5 + n = 2.291287847

Solving
0.5 + n = 2.291287847

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 = 2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 2.291287847 + -0.5
n = 2.291287847 + -0.5

Combine like terms: 2.291287847 + -0.5 = 1.791287847
n = 1.791287847

Simplifying
n = 1.791287847


Subproblem 2
n + 0.5 = -2.291287847

Simplifying
n + 0.5 = -2.291287847

Reorder the terms:
0.5 + n = -2.291287847

Solving
0.5 + n = -2.291287847

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 = -2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -2.291287847 + -0.5
n = -2.291287847 + -0.5

Combine like terms: -2.291287847 + -0.5 = -2.791287847
n = -2.791287847

Simplifying
n = -2.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.791287847, -2.791287847}

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