n(n-7)=90

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Solution for n(n-7)=90 equation: 


Simplifying
n(n + -7) = 90

Reorder the terms:
n(-7 + n) = 90
(-7 * n + n * n) = 90
(-7n + n2) = 90

Solving
-7n + n2 = 90

Solving for variable 'n'.

Reorder the terms:
-90 + -7n + n2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + -7n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + -7n + 90 + n2 = 0 + 90

Reorder the terms:
-90 + 90 + -7n + n2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + -7n + n2 = 0 + 90
-7n + n2 = 0 + 90

Combine like terms: 0 + 90 = 90
-7n + n2 = 90

The n term is -7n.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7n + 12.25 + n2 = 90 + 12.25

Reorder the terms:
12.25 + -7n + n2 = 90 + 12.25

Combine like terms: 90 + 12.25 = 102.25
12.25 + -7n + n2 = 102.25

Factor a perfect square on the left side:
(n + -3.5)(n + -3.5) = 102.25

Calculate the square root of the right side: 10.111874208

Break this problem into two subproblems by setting 
(n + -3.5) equal to 10.111874208 and -10.111874208.


Subproblem 1
n + -3.5 = 10.111874208

Simplifying
n + -3.5 = 10.111874208

Reorder the terms:
-3.5 + n = 10.111874208

Solving
-3.5 + n = 10.111874208

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = 10.111874208 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = 10.111874208 + 3.5
n = 10.111874208 + 3.5

Combine like terms: 10.111874208 + 3.5 = 13.611874208
n = 13.611874208

Simplifying
n = 13.611874208


Subproblem 2
n + -3.5 = -10.111874208

Simplifying
n + -3.5 = -10.111874208

Reorder the terms:
-3.5 + n = -10.111874208

Solving
-3.5 + n = -10.111874208

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n = -10.111874208 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n = -10.111874208 + 3.5
n = -10.111874208 + 3.5

Combine like terms: -10.111874208 + 3.5 = -6.611874208
n = -6.611874208

Simplifying
n = -6.611874208


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {13.611874208, -6.611874208}

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