n^2-n-500=0

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


Simplifying
n2 + -1n + -500 = 0

Reorder the terms:
-500 + -1n + n2 = 0

Solving
-500 + -1n + 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 + -1n + 500 + n2 = 0 + 500

Reorder the terms:
-500 + 500 + -1n + n2 = 0 + 500

Combine like terms: -500 + 500 = 0
0 + -1n + n2 = 0 + 500
-1n + n2 = 0 + 500

Combine like terms: 0 + 500 = 500
-1n + n2 = 500

The n term is -1n.  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.
-1n + 0.25 + n2 = 500 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 500 + 0.25

Combine like terms: 500 + 0.25 = 500.25
0.25 + -1n + n2 = 500.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 500.25

Calculate the square root of the right side: 22.366269246

Break this problem into two subproblems by setting 
(n + -0.5) equal to 22.366269246 and -22.366269246.

Subproblem 1

n + -0.5 = 22.366269246 Simplifying n + -0.5 = 22.366269246 Reorder the terms: -0.5 + n = 22.366269246 Solving -0.5 + n = 22.366269246 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 = 22.366269246 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 22.366269246 + 0.5 n = 22.366269246 + 0.5 Combine like terms: 22.366269246 + 0.5 = 22.866269246 n = 22.866269246 Simplifying n = 22.866269246

Subproblem 2

n + -0.5 = -22.366269246 Simplifying n + -0.5 = -22.366269246 Reorder the terms: -0.5 + n = -22.366269246 Solving -0.5 + n = -22.366269246 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 = -22.366269246 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -22.366269246 + 0.5 n = -22.366269246 + 0.5 Combine like terms: -22.366269246 + 0.5 = -21.866269246 n = -21.866269246 Simplifying n = -21.866269246

Solution

The solution to the problem is based on the solutions from the subproblems. n = {22.866269246, -21.866269246}

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