n^2-3n-65=0

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Solution for n^2-3n-65=0 equation: 


Simplifying
n2 + -3n + -65 = 0

Reorder the terms:
-65 + -3n + n2 = 0

Solving
-65 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '65' to each side of the equation.
-65 + -3n + 65 + n2 = 0 + 65

Reorder the terms:
-65 + 65 + -3n + n2 = 0 + 65

Combine like terms: -65 + 65 = 0
0 + -3n + n2 = 0 + 65
-3n + n2 = 0 + 65

Combine like terms: 0 + 65 = 65
-3n + n2 = 65

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 = 65 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 65 + 2.25

Combine like terms: 65 + 2.25 = 67.25
2.25 + -3n + n2 = 67.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 67.25

Calculate the square root of the right side: 8.200609733

Break this problem into two subproblems by setting 
(n + -1.5) equal to 8.200609733 and -8.200609733.


Subproblem 1
n + -1.5 = 8.200609733

Simplifying
n + -1.5 = 8.200609733

Reorder the terms:
-1.5 + n = 8.200609733

Solving
-1.5 + n = 8.200609733

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 = 8.200609733 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 8.200609733 + 1.5
n = 8.200609733 + 1.5

Combine like terms: 8.200609733 + 1.5 = 9.700609733
n = 9.700609733

Simplifying
n = 9.700609733


Subproblem 2
n + -1.5 = -8.200609733

Simplifying
n + -1.5 = -8.200609733

Reorder the terms:
-1.5 + n = -8.200609733

Solving
-1.5 + n = -8.200609733

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 = -8.200609733 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -8.200609733 + 1.5
n = -8.200609733 + 1.5

Combine like terms: -8.200609733 + 1.5 = -6.700609733
n = -6.700609733

Simplifying
n = -6.700609733


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {9.700609733, -6.700609733}

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