n^2-2n-52=0

Solution for n^2-2n-52=0 equation:

Simplifying
n2 + -2n + -52 = 0

Reorder the terms:
-52 + -2n + n2 = 0

Solving
-52 + -2n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '52' to each side of the equation.
-52 + -2n + 52 + n2 = 0 + 52

Reorder the terms:
-52 + 52 + -2n + n2 = 0 + 52

Combine like terms: -52 + 52 = 0
0 + -2n + n2 = 0 + 52
-2n + n2 = 0 + 52

Combine like terms: 0 + 52 = 52
-2n + n2 = 52

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 52 + 1

Reorder the terms:
1 + -2n + n2 = 52 + 1

Combine like terms: 52 + 1 = 53
1 + -2n + n2 = 53

Factor a perfect square on the left side:
(n + -1)(n + -1) = 53

Calculate the square root of the right side: 7.280109889

Break this problem into two subproblems by setting 
(n + -1) equal to 7.280109889 and -7.280109889.

Subproblem 1
n + -1 = 7.280109889

Simplifying
n + -1 = 7.280109889

Reorder the terms:
-1 + n = 7.280109889

Solving
-1 + n = 7.280109889

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = 7.280109889 + 1

Combine like terms: -1 + 1 = 0
0 + n = 7.280109889 + 1
n = 7.280109889 + 1

Combine like terms: 7.280109889 + 1 = 8.280109889
n = 8.280109889

Simplifying
n = 8.280109889

Subproblem 2
n + -1 = -7.280109889

Simplifying
n + -1 = -7.280109889

Reorder the terms:
-1 + n = -7.280109889

Solving
-1 + n = -7.280109889

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = -7.280109889 + 1

Combine like terms: -1 + 1 = 0
0 + n = -7.280109889 + 1
n = -7.280109889 + 1

Combine like terms: -7.280109889 + 1 = -6.280109889
n = -6.280109889

Simplifying
n = -6.280109889

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {8.280109889, -6.280109889}