n^2-3n-61=0

Solution for n^2-3n-61=0 equation:

Simplifying
n2 + -3n + -61 = 0

Reorder the terms:
-61 + -3n + n2 = 0

Solving
-61 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '61' to each side of the equation.
-61 + -3n + 61 + n2 = 0 + 61

Reorder the terms:
-61 + 61 + -3n + n2 = 0 + 61

Combine like terms: -61 + 61 = 0
0 + -3n + n2 = 0 + 61
-3n + n2 = 0 + 61

Combine like terms: 0 + 61 = 61
-3n + n2 = 61

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 = 61 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 61 + 2.25

Combine like terms: 61 + 2.25 = 63.25
2.25 + -3n + n2 = 63.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 63.25

Calculate the square root of the right side: 7.95298686

Break this problem into two subproblems by setting 
(n + -1.5) equal to 7.95298686 and -7.95298686.

Subproblem 1
n + -1.5 = 7.95298686

Simplifying
n + -1.5 = 7.95298686

Reorder the terms:
-1.5 + n = 7.95298686

Solving
-1.5 + n = 7.95298686

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 = 7.95298686 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 7.95298686 + 1.5
n = 7.95298686 + 1.5

Combine like terms: 7.95298686 + 1.5 = 9.45298686
n = 9.45298686

Simplifying
n = 9.45298686

Subproblem 2
n + -1.5 = -7.95298686

Simplifying
n + -1.5 = -7.95298686

Reorder the terms:
-1.5 + n = -7.95298686

Solving
-1.5 + n = -7.95298686

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 = -7.95298686 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -7.95298686 + 1.5
n = -7.95298686 + 1.5

Combine like terms: -7.95298686 + 1.5 = -6.45298686
n = -6.45298686

Simplifying
n = -6.45298686

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {9.45298686, -6.45298686}