n^2+n=198

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Solution for n^2+n=198 equation:


Simplifying
n2 + n = 198

Reorder the terms:
n + n2 = 198

Solving
n + n2 = 198

Solving for variable 'n'.

Reorder the terms:
-198 + n + n2 = 198 + -198

Combine like terms: 198 + -198 = 0
-198 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '198' to each side of the equation.
-198 + n + 198 + n2 = 0 + 198

Reorder the terms:
-198 + 198 + n + n2 = 0 + 198

Combine like terms: -198 + 198 = 0
0 + n + n2 = 0 + 198
n + n2 = 0 + 198

Combine like terms: 0 + 198 = 198
n + n2 = 198

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 = 198 + 0.25

Reorder the terms:
0.25 + n + n2 = 198 + 0.25

Combine like terms: 198 + 0.25 = 198.25
0.25 + n + n2 = 198.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 198.25

Calculate the square root of the right side: 14.08012784

Break this problem into two subproblems by setting 
(n + 0.5) equal to 14.08012784 and -14.08012784.

Subproblem 1

n + 0.5 = 14.08012784 Simplifying n + 0.5 = 14.08012784 Reorder the terms: 0.5 + n = 14.08012784 Solving 0.5 + n = 14.08012784 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 = 14.08012784 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 14.08012784 + -0.5 n = 14.08012784 + -0.5 Combine like terms: 14.08012784 + -0.5 = 13.58012784 n = 13.58012784 Simplifying n = 13.58012784

Subproblem 2

n + 0.5 = -14.08012784 Simplifying n + 0.5 = -14.08012784 Reorder the terms: 0.5 + n = -14.08012784 Solving 0.5 + n = -14.08012784 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 = -14.08012784 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -14.08012784 + -0.5 n = -14.08012784 + -0.5 Combine like terms: -14.08012784 + -0.5 = -14.58012784 n = -14.58012784 Simplifying n = -14.58012784

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.58012784, -14.58012784}

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