n^2+23n-500=0

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Solution for n^2+23n-500=0 equation: 


Simplifying
n2 + 23n + -500 = 0

Reorder the terms:
-500 + 23n + n2 = 0

Solving
-500 + 23n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '500' to each side of the equation.
-500 + 23n + 500 + n2 = 0 + 500

Reorder the terms:
-500 + 500 + 23n + n2 = 0 + 500

Combine like terms: -500 + 500 = 0
0 + 23n + n2 = 0 + 500
23n + n2 = 0 + 500

Combine like terms: 0 + 500 = 500
23n + n2 = 500

The n term is 23n.  Take half its coefficient (11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
23n + 132.25 + n2 = 500 + 132.25

Reorder the terms:
132.25 + 23n + n2 = 500 + 132.25

Combine like terms: 500 + 132.25 = 632.25
132.25 + 23n + n2 = 632.25

Factor a perfect square on the left side:
(n + 11.5)(n + 11.5) = 632.25

Calculate the square root of the right side: 25.144581921

Break this problem into two subproblems by setting 
(n + 11.5) equal to 25.144581921 and -25.144581921.


Subproblem 1
n + 11.5 = 25.144581921

Simplifying
n + 11.5 = 25.144581921

Reorder the terms:
11.5 + n = 25.144581921

Solving
11.5 + n = 25.144581921

Solving for variable 'n'.

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

Add '-11.5' to each side of the equation.
11.5 + -11.5 + n = 25.144581921 + -11.5

Combine like terms: 11.5 + -11.5 = 0.0
0.0 + n = 25.144581921 + -11.5
n = 25.144581921 + -11.5

Combine like terms: 25.144581921 + -11.5 = 13.644581921
n = 13.644581921

Simplifying
n = 13.644581921


Subproblem 2
n + 11.5 = -25.144581921

Simplifying
n + 11.5 = -25.144581921

Reorder the terms:
11.5 + n = -25.144581921

Solving
11.5 + n = -25.144581921

Solving for variable 'n'.

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

Add '-11.5' to each side of the equation.
11.5 + -11.5 + n = -25.144581921 + -11.5

Combine like terms: 11.5 + -11.5 = 0.0
0.0 + n = -25.144581921 + -11.5
n = -25.144581921 + -11.5

Combine like terms: -25.144581921 + -11.5 = -36.644581921
n = -36.644581921

Simplifying
n = -36.644581921


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {13.644581921, -36.644581921}

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