N^2+181.125N-54337.5=0

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Solution for N^2+181.125N-54337.5=0 equation: 


Simplifying
N2 + 181.125N + -54337.5 = 0

Reorder the terms:
-54337.5 + 181.125N + N2 = 0

Solving
-54337.5 + 181.125N + N2 = 0

Solving for variable 'N'.

Begin completing the square.

Move the constant term to the right:

Add '54337.5' to each side of the equation.
-54337.5 + 181.125N + 54337.5 + N2 = 0 + 54337.5

Reorder the terms:
-54337.5 + 54337.5 + 181.125N + N2 = 0 + 54337.5

Combine like terms: -54337.5 + 54337.5 = 0.0
0.0 + 181.125N + N2 = 0 + 54337.5
181.125N + N2 = 0 + 54337.5

Combine like terms: 0 + 54337.5 = 54337.5
181.125N + N2 = 54337.5

The N term is 181.125N.  Take half its coefficient (90.5625).
Square it (8201.566406) and add it to both sides.

Add '8201.566406' to each side of the equation.
181.125N + 8201.566406 + N2 = 54337.5 + 8201.566406

Reorder the terms:
8201.566406 + 181.125N + N2 = 54337.5 + 8201.566406

Combine like terms: 54337.5 + 8201.566406 = 62539.066406
8201.566406 + 181.125N + N2 = 62539.066406

Factor a perfect square on the left side:
(N + 90.5625)(N + 90.5625) = 62539.066406

Calculate the square root of the right side: 250.078120606

Break this problem into two subproblems by setting 
(N + 90.5625) equal to 250.078120606 and -250.078120606.


Subproblem 1
N + 90.5625 = 250.078120606

Simplifying
N + 90.5625 = 250.078120606

Reorder the terms:
90.5625 + N = 250.078120606

Solving
90.5625 + N = 250.078120606

Solving for variable 'N'.

Move all terms containing N to the left, all other terms to the right.

Add '-90.5625' to each side of the equation.
90.5625 + -90.5625 + N = 250.078120606 + -90.5625

Combine like terms: 90.5625 + -90.5625 = 0.0000
0.0000 + N = 250.078120606 + -90.5625
N = 250.078120606 + -90.5625

Combine like terms: 250.078120606 + -90.5625 = 159.515620606
N = 159.515620606

Simplifying
N = 159.515620606


Subproblem 2
N + 90.5625 = -250.078120606

Simplifying
N + 90.5625 = -250.078120606

Reorder the terms:
90.5625 + N = -250.078120606

Solving
90.5625 + N = -250.078120606

Solving for variable 'N'.

Move all terms containing N to the left, all other terms to the right.

Add '-90.5625' to each side of the equation.
90.5625 + -90.5625 + N = -250.078120606 + -90.5625

Combine like terms: 90.5625 + -90.5625 = 0.0000
0.0000 + N = -250.078120606 + -90.5625
N = -250.078120606 + -90.5625

Combine like terms: -250.078120606 + -90.5625 = -340.640620606
N = -340.640620606

Simplifying
N = -340.640620606


Solution
The solution to the problem is based on the solutions
from the subproblems.
N = {159.515620606, -340.640620606}

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