n^2-45n=116

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Solution for n^2-45n=116 equation:


Simplifying
n2 + -45n = 116

Reorder the terms:
-45n + n2 = 116

Solving
-45n + n2 = 116

Solving for variable 'n'.

Reorder the terms:
-116 + -45n + n2 = 116 + -116

Combine like terms: 116 + -116 = 0
-116 + -45n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '116' to each side of the equation.
-116 + -45n + 116 + n2 = 0 + 116

Reorder the terms:
-116 + 116 + -45n + n2 = 0 + 116

Combine like terms: -116 + 116 = 0
0 + -45n + n2 = 0 + 116
-45n + n2 = 0 + 116

Combine like terms: 0 + 116 = 116
-45n + n2 = 116

The n term is -45n.  Take half its coefficient (-22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
-45n + 506.25 + n2 = 116 + 506.25

Reorder the terms:
506.25 + -45n + n2 = 116 + 506.25

Combine like terms: 116 + 506.25 = 622.25
506.25 + -45n + n2 = 622.25

Factor a perfect square on the left side:
(n + -22.5)(n + -22.5) = 622.25

Calculate the square root of the right side: 24.944939367

Break this problem into two subproblems by setting 
(n + -22.5) equal to 24.944939367 and -24.944939367.

Subproblem 1

n + -22.5 = 24.944939367 Simplifying n + -22.5 = 24.944939367 Reorder the terms: -22.5 + n = 24.944939367 Solving -22.5 + n = 24.944939367 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + n = 24.944939367 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + n = 24.944939367 + 22.5 n = 24.944939367 + 22.5 Combine like terms: 24.944939367 + 22.5 = 47.444939367 n = 47.444939367 Simplifying n = 47.444939367

Subproblem 2

n + -22.5 = -24.944939367 Simplifying n + -22.5 = -24.944939367 Reorder the terms: -22.5 + n = -24.944939367 Solving -22.5 + n = -24.944939367 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + n = -24.944939367 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + n = -24.944939367 + 22.5 n = -24.944939367 + 22.5 Combine like terms: -24.944939367 + 22.5 = -2.444939367 n = -2.444939367 Simplifying n = -2.444939367

Solution

The solution to the problem is based on the solutions from the subproblems. n = {47.444939367, -2.444939367}

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