n^2-3n=314.5

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Solution for n^2-3n=314.5 equation: 


Simplifying
n2 + -3n = 314.5

Reorder the terms:
-3n + n2 = 314.5

Solving
-3n + n2 = 314.5

Solving for variable 'n'.

Reorder the terms:
-314.5 + -3n + n2 = 314.5 + -314.5

Combine like terms: 314.5 + -314.5 = 0.0
-314.5 + -3n + n2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '314.5' to each side of the equation.
-314.5 + -3n + 314.5 + n2 = 0.0 + 314.5

Reorder the terms:
-314.5 + 314.5 + -3n + n2 = 0.0 + 314.5

Combine like terms: -314.5 + 314.5 = 0.0
0.0 + -3n + n2 = 0.0 + 314.5
-3n + n2 = 0.0 + 314.5

Combine like terms: 0.0 + 314.5 = 314.5
-3n + n2 = 314.5

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 314.5 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 314.5 + 2.25

Combine like terms: 314.5 + 2.25 = 316.75
2.25 + -3n + n2 = 316.75

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 316.75

Calculate the square root of the right side: 17.797471731

Break this problem into two subproblems by setting 
(n + -1.5) equal to 17.797471731 and -17.797471731.


Subproblem 1
n + -1.5 = 17.797471731

Simplifying
n + -1.5 = 17.797471731

Reorder the terms:
-1.5 + n = 17.797471731

Solving
-1.5 + n = 17.797471731

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = 17.797471731 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 17.797471731 + 1.5
n = 17.797471731 + 1.5

Combine like terms: 17.797471731 + 1.5 = 19.297471731
n = 19.297471731

Simplifying
n = 19.297471731


Subproblem 2
n + -1.5 = -17.797471731

Simplifying
n + -1.5 = -17.797471731

Reorder the terms:
-1.5 + n = -17.797471731

Solving
-1.5 + n = -17.797471731

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = -17.797471731 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -17.797471731 + 1.5
n = -17.797471731 + 1.5

Combine like terms: -17.797471731 + 1.5 = -16.297471731
n = -16.297471731

Simplifying
n = -16.297471731


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {19.297471731, -16.297471731}

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