n^2+n-1000=0

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Solution for n^2+n-1000=0 equation:


Simplifying
n2 + n + -1000 = 0

Reorder the terms:
-1000 + n + n2 = 0

Solving
-1000 + n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '1000' to each side of the equation.
-1000 + n + 1000 + n2 = 0 + 1000

Reorder the terms:
-1000 + 1000 + n + n2 = 0 + 1000

Combine like terms: -1000 + 1000 = 0
0 + n + n2 = 0 + 1000
n + n2 = 0 + 1000

Combine like terms: 0 + 1000 = 1000
n + n2 = 1000

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 = 1000 + 0.25

Reorder the terms:
0.25 + n + n2 = 1000 + 0.25

Combine like terms: 1000 + 0.25 = 1000.25
0.25 + n + n2 = 1000.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 1000.25

Calculate the square root of the right side: 31.626729202

Break this problem into two subproblems by setting 
(n + 0.5) equal to 31.626729202 and -31.626729202.

Subproblem 1

n + 0.5 = 31.626729202 Simplifying n + 0.5 = 31.626729202 Reorder the terms: 0.5 + n = 31.626729202 Solving 0.5 + n = 31.626729202 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 = 31.626729202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 31.626729202 + -0.5 n = 31.626729202 + -0.5 Combine like terms: 31.626729202 + -0.5 = 31.126729202 n = 31.126729202 Simplifying n = 31.126729202

Subproblem 2

n + 0.5 = -31.626729202 Simplifying n + 0.5 = -31.626729202 Reorder the terms: 0.5 + n = -31.626729202 Solving 0.5 + n = -31.626729202 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 = -31.626729202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -31.626729202 + -0.5 n = -31.626729202 + -0.5 Combine like terms: -31.626729202 + -0.5 = -32.126729202 n = -32.126729202 Simplifying n = -32.126729202

Solution

The solution to the problem is based on the solutions from the subproblems. n = {31.126729202, -32.126729202}

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