n(n-1)=90.33

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Solution for n(n-1)=90.33 equation: 


Simplifying
n(n + -1) = 90.33

Reorder the terms:
n(-1 + n) = 90.33
(-1 * n + n * n) = 90.33
(-1n + n2) = 90.33

Solving
-1n + n2 = 90.33

Solving for variable 'n'.

Reorder the terms:
-90.33 + -1n + n2 = 90.33 + -90.33

Combine like terms: 90.33 + -90.33 = 0.00
-90.33 + -1n + n2 = 0.00

Begin completing the square.

Move the constant term to the right:

Add '90.33' to each side of the equation.
-90.33 + -1n + 90.33 + n2 = 0.00 + 90.33

Reorder the terms:
-90.33 + 90.33 + -1n + n2 = 0.00 + 90.33

Combine like terms: -90.33 + 90.33 = 0.00
0.00 + -1n + n2 = 0.00 + 90.33
-1n + n2 = 0.00 + 90.33

Combine like terms: 0.00 + 90.33 = 90.33
-1n + n2 = 90.33

The n term is -1n.  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.
-1n + 0.25 + n2 = 90.33 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 90.33 + 0.25

Combine like terms: 90.33 + 0.25 = 90.58
0.25 + -1n + n2 = 90.58

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 90.58

Calculate the square root of the right side: 9.517352573

Break this problem into two subproblems by setting 
(n + -0.5) equal to 9.517352573 and -9.517352573.


Subproblem 1
n + -0.5 = 9.517352573

Simplifying
n + -0.5 = 9.517352573

Reorder the terms:
-0.5 + n = 9.517352573

Solving
-0.5 + n = 9.517352573

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 = 9.517352573 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 9.517352573 + 0.5
n = 9.517352573 + 0.5

Combine like terms: 9.517352573 + 0.5 = 10.017352573
n = 10.017352573

Simplifying
n = 10.017352573


Subproblem 2
n + -0.5 = -9.517352573

Simplifying
n + -0.5 = -9.517352573

Reorder the terms:
-0.5 + n = -9.517352573

Solving
-0.5 + n = -9.517352573

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 = -9.517352573 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -9.517352573 + 0.5
n = -9.517352573 + 0.5

Combine like terms: -9.517352573 + 0.5 = -9.017352573
n = -9.017352573

Simplifying
n = -9.017352573


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {10.017352573, -9.017352573}

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