n^2-4n-61=0

Solution for n^2-4n-61=0 equation:

Simplifying
n2 + -4n + -61 = 0

Reorder the terms:
-61 + -4n + n2 = 0

Solving
-61 + -4n + 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 + -4n + 61 + n2 = 0 + 61

Reorder the terms:
-61 + 61 + -4n + n2 = 0 + 61

Combine like terms: -61 + 61 = 0
0 + -4n + n2 = 0 + 61
-4n + n2 = 0 + 61

Combine like terms: 0 + 61 = 61
-4n + n2 = 61

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 61 + 4

Reorder the terms:
4 + -4n + n2 = 61 + 4

Combine like terms: 61 + 4 = 65
4 + -4n + n2 = 65

Factor a perfect square on the left side:
(n + -2)(n + -2) = 65

Calculate the square root of the right side: 8.062257748

Break this problem into two subproblems by setting 
(n + -2) equal to 8.062257748 and -8.062257748.

Subproblem 1
n + -2 = 8.062257748

Simplifying
n + -2 = 8.062257748

Reorder the terms:
-2 + n = 8.062257748

Solving
-2 + n = 8.062257748

Solving for variable 'n'.

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

Add '2' to each side of the equation.
-2 + 2 + n = 8.062257748 + 2

Combine like terms: -2 + 2 = 0
0 + n = 8.062257748 + 2
n = 8.062257748 + 2

Combine like terms: 8.062257748 + 2 = 10.062257748
n = 10.062257748

Simplifying
n = 10.062257748

Subproblem 2
n + -2 = -8.062257748

Simplifying
n + -2 = -8.062257748

Reorder the terms:
-2 + n = -8.062257748

Solving
-2 + n = -8.062257748

Solving for variable 'n'.

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

Add '2' to each side of the equation.
-2 + 2 + n = -8.062257748 + 2

Combine like terms: -2 + 2 = 0
0 + n = -8.062257748 + 2
n = -8.062257748 + 2

Combine like terms: -8.062257748 + 2 = -6.062257748
n = -6.062257748

Simplifying
n = -6.062257748

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {10.062257748, -6.062257748}