5n^2-12n=-1

Solution for 5n^2-12n=-1 equation:

Simplifying
5n2 + -12n = -1

Reorder the terms:
-12n + 5n2 = -1

Solving
-12n + 5n2 = -1

Solving for variable 'n'.

Reorder the terms:
1 + -12n + 5n2 = -1 + 1

Combine like terms: -1 + 1 = 0
1 + -12n + 5n2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
0.2 + -2.4n + n2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -2.4n + -0.2 + n2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -2.4n + n2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -2.4n + n2 = 0 + -0.2
-2.4n + n2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-2.4n + n2 = -0.2

The n term is -2.4n.  Take half its coefficient (-1.2).
Square it (1.44) and add it to both sides.

Add '1.44' to each side of the equation.
-2.4n + 1.44 + n2 = -0.2 + 1.44

Reorder the terms:
1.44 + -2.4n + n2 = -0.2 + 1.44

Combine like terms: -0.2 + 1.44 = 1.24
1.44 + -2.4n + n2 = 1.24

Factor a perfect square on the left side:
(n + -1.2)(n + -1.2) = 1.24

Calculate the square root of the right side: 1.113552873

Break this problem into two subproblems by setting 
(n + -1.2) equal to 1.113552873 and -1.113552873.

Subproblem 1
n + -1.2 = 1.113552873

Simplifying
n + -1.2 = 1.113552873

Reorder the terms:
-1.2 + n = 1.113552873

Solving
-1.2 + n = 1.113552873

Solving for variable 'n'.

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

Add '1.2' to each side of the equation.
-1.2 + 1.2 + n = 1.113552873 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + n = 1.113552873 + 1.2
n = 1.113552873 + 1.2

Combine like terms: 1.113552873 + 1.2 = 2.313552873
n = 2.313552873

Simplifying
n = 2.313552873

Subproblem 2
n + -1.2 = -1.113552873

Simplifying
n + -1.2 = -1.113552873

Reorder the terms:
-1.2 + n = -1.113552873

Solving
-1.2 + n = -1.113552873

Solving for variable 'n'.

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

Add '1.2' to each side of the equation.
-1.2 + 1.2 + n = -1.113552873 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + n = -1.113552873 + 1.2
n = -1.113552873 + 1.2

Combine like terms: -1.113552873 + 1.2 = 0.086447127
n = 0.086447127

Simplifying
n = 0.086447127

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.313552873, 0.086447127}