n^2+n-512=0

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Solution for n^2+n-512=0 equation:


Simplifying
n2 + n + -512 = 0

Reorder the terms:
-512 + n + n2 = 0

Solving
-512 + n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '512' to each side of the equation.
-512 + n + 512 + n2 = 0 + 512

Reorder the terms:
-512 + 512 + n + n2 = 0 + 512

Combine like terms: -512 + 512 = 0
0 + n + n2 = 0 + 512
n + n2 = 0 + 512

Combine like terms: 0 + 512 = 512
n + n2 = 512

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 = 512 + 0.25

Reorder the terms:
0.25 + n + n2 = 512 + 0.25

Combine like terms: 512 + 0.25 = 512.25
0.25 + n + n2 = 512.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 512.25

Calculate the square root of the right side: 22.632940596

Break this problem into two subproblems by setting 
(n + 0.5) equal to 22.632940596 and -22.632940596.

Subproblem 1

n + 0.5 = 22.632940596 Simplifying n + 0.5 = 22.632940596 Reorder the terms: 0.5 + n = 22.632940596 Solving 0.5 + n = 22.632940596 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 = 22.632940596 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 22.632940596 + -0.5 n = 22.632940596 + -0.5 Combine like terms: 22.632940596 + -0.5 = 22.132940596 n = 22.132940596 Simplifying n = 22.132940596

Subproblem 2

n + 0.5 = -22.632940596 Simplifying n + 0.5 = -22.632940596 Reorder the terms: 0.5 + n = -22.632940596 Solving 0.5 + n = -22.632940596 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 = -22.632940596 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -22.632940596 + -0.5 n = -22.632940596 + -0.5 Combine like terms: -22.632940596 + -0.5 = -23.132940596 n = -23.132940596 Simplifying n = -23.132940596

Solution

The solution to the problem is based on the solutions from the subproblems. n = {22.132940596, -23.132940596}

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