n^2+n=20000

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Solution for n^2+n=20000 equation:


Simplifying
n2 + n = 20000

Reorder the terms:
n + n2 = 20000

Solving
n + n2 = 20000

Solving for variable 'n'.

Reorder the terms:
-20000 + n + n2 = 20000 + -20000

Combine like terms: 20000 + -20000 = 0
-20000 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '20000' to each side of the equation.
-20000 + n + 20000 + n2 = 0 + 20000

Reorder the terms:
-20000 + 20000 + n + n2 = 0 + 20000

Combine like terms: -20000 + 20000 = 0
0 + n + n2 = 0 + 20000
n + n2 = 0 + 20000

Combine like terms: 0 + 20000 = 20000
n + n2 = 20000

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 = 20000 + 0.25

Reorder the terms:
0.25 + n + n2 = 20000 + 0.25

Combine like terms: 20000 + 0.25 = 20000.25
0.25 + n + n2 = 20000.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 20000.25

Calculate the square root of the right side: 141.422240118

Break this problem into two subproblems by setting 
(n + 0.5) equal to 141.422240118 and -141.422240118.

Subproblem 1

n + 0.5 = 141.422240118 Simplifying n + 0.5 = 141.422240118 Reorder the terms: 0.5 + n = 141.422240118 Solving 0.5 + n = 141.422240118 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 = 141.422240118 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 141.422240118 + -0.5 n = 141.422240118 + -0.5 Combine like terms: 141.422240118 + -0.5 = 140.922240118 n = 140.922240118 Simplifying n = 140.922240118

Subproblem 2

n + 0.5 = -141.422240118 Simplifying n + 0.5 = -141.422240118 Reorder the terms: 0.5 + n = -141.422240118 Solving 0.5 + n = -141.422240118 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 = -141.422240118 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -141.422240118 + -0.5 n = -141.422240118 + -0.5 Combine like terms: -141.422240118 + -0.5 = -141.922240118 n = -141.922240118 Simplifying n = -141.922240118

Solution

The solution to the problem is based on the solutions from the subproblems. n = {140.922240118, -141.922240118}

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