(5-6y)(5-6y)=113

Solution for (5-6y)(5-6y)=113 equation:

Simplifying
(5 + -6y)(5 + -6y) = 113

Multiply (5 + -6y) * (5 + -6y)
(5(5 + -6y) + -6y * (5 + -6y)) = 113
((5 * 5 + -6y * 5) + -6y * (5 + -6y)) = 113
((25 + -30y) + -6y * (5 + -6y)) = 113
(25 + -30y + (5 * -6y + -6y * -6y)) = 113
(25 + -30y + (-30y + 36y2)) = 113

Combine like terms: -30y + -30y = -60y
(25 + -60y + 36y2) = 113

Solving
25 + -60y + 36y2 = 113

Solving for variable 'y'.

Reorder the terms:
25 + -113 + -60y + 36y2 = 113 + -113

Combine like terms: 25 + -113 = -88
-88 + -60y + 36y2 = 113 + -113

Combine like terms: 113 + -113 = 0
-88 + -60y + 36y2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-22 + -15y + 9y2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-22 + -15y + 9y2)' equal to zero and attempt to solve:

Simplifying
-22 + -15y + 9y2 = 0

Solving
-22 + -15y + 9y2 = 0

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

Divide each side by '9'.
-2.444444444 + -1.666666667y + y2 = 0

Move the constant term to the right:

Add '2.444444444' to each side of the equation.
-2.444444444 + -1.666666667y + 2.444444444 + y2 = 0 + 2.444444444

Reorder the terms:
-2.444444444 + 2.444444444 + -1.666666667y + y2 = 0 + 2.444444444

Combine like terms: -2.444444444 + 2.444444444 = 0.000000000
0.000000000 + -1.666666667y + y2 = 0 + 2.444444444
-1.666666667y + y2 = 0 + 2.444444444

Combine like terms: 0 + 2.444444444 = 2.444444444
-1.666666667y + y2 = 2.444444444

The y term is -1.666666667y.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667y + 0.6944444447 + y2 = 2.444444444 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667y + y2 = 2.444444444 + 0.6944444447

Combine like terms: 2.444444444 + 0.6944444447 = 3.1388888887
0.6944444447 + -1.666666667y + y2 = 3.1388888887

Factor a perfect square on the left side:
(y + -0.8333333335)(y + -0.8333333335) = 3.1388888887

Calculate the square root of the right side: 1.771690969

Break this problem into two subproblems by setting 
(y + -0.8333333335) equal to 1.771690969 and -1.771690969.

Subproblem 1
y + -0.8333333335 = 1.771690969

Simplifying
y + -0.8333333335 = 1.771690969

Reorder the terms:
-0.8333333335 + y = 1.771690969

Solving
-0.8333333335 + y = 1.771690969

Solving for variable 'y'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + y = 1.771690969 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + y = 1.771690969 + 0.8333333335
y = 1.771690969 + 0.8333333335

Combine like terms: 1.771690969 + 0.8333333335 = 2.6050243025
y = 2.6050243025

Simplifying
y = 2.6050243025

Subproblem 2
y + -0.8333333335 = -1.771690969

Simplifying
y + -0.8333333335 = -1.771690969

Reorder the terms:
-0.8333333335 + y = -1.771690969

Solving
-0.8333333335 + y = -1.771690969

Solving for variable 'y'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + y = -1.771690969 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + y = -1.771690969 + 0.8333333335
y = -1.771690969 + 0.8333333335

Combine like terms: -1.771690969 + 0.8333333335 = -0.9383576355
y = -0.9383576355

Simplifying
y = -0.9383576355

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.6050243025, -0.9383576355}
Solution
y = {2.6050243025, -0.9383576355}