(y-1)2+(2y+3)2=5(y+1)(y-1)

Solution for (y-1)2+(2y+3)2=5(y+1)(y-1) equation:

Simplifying
(y + -1) * 2 + (2y + 3) * 2 = 5(y + 1)(y + -1)

Reorder the terms:
(-1 + y) * 2 + (2y + 3) * 2 = 5(y + 1)(y + -1)

Reorder the terms for easier multiplication:
2(-1 + y) + (2y + 3) * 2 = 5(y + 1)(y + -1)
(-1 * 2 + y * 2) + (2y + 3) * 2 = 5(y + 1)(y + -1)
(-2 + 2y) + (2y + 3) * 2 = 5(y + 1)(y + -1)

Reorder the terms:
-2 + 2y + (3 + 2y) * 2 = 5(y + 1)(y + -1)

Reorder the terms for easier multiplication:
-2 + 2y + 2(3 + 2y) = 5(y + 1)(y + -1)
-2 + 2y + (3 * 2 + 2y * 2) = 5(y + 1)(y + -1)
-2 + 2y + (6 + 4y) = 5(y + 1)(y + -1)

Reorder the terms:
-2 + 6 + 2y + 4y = 5(y + 1)(y + -1)

Combine like terms: -2 + 6 = 4
4 + 2y + 4y = 5(y + 1)(y + -1)

Combine like terms: 2y + 4y = 6y
4 + 6y = 5(y + 1)(y + -1)

Reorder the terms:
4 + 6y = 5(1 + y)(y + -1)

Reorder the terms:
4 + 6y = 5(1 + y)(-1 + y)

Multiply (1 + y) * (-1 + y)
4 + 6y = 5(1(-1 + y) + y(-1 + y))
4 + 6y = 5((-1 * 1 + y * 1) + y(-1 + y))
4 + 6y = 5((-1 + 1y) + y(-1 + y))
4 + 6y = 5(-1 + 1y + (-1 * y + y * y))
4 + 6y = 5(-1 + 1y + (-1y + y2))

Combine like terms: 1y + -1y = 0
4 + 6y = 5(-1 + 0 + y2)
4 + 6y = 5(-1 + y2)
4 + 6y = (-1 * 5 + y2 * 5)
4 + 6y = (-5 + 5y2)

Solving
4 + 6y = -5 + 5y2

Solving for variable 'y'.

Reorder the terms:
4 + 5 + 6y + -5y2 = -5 + 5y2 + 5 + -5y2

Combine like terms: 4 + 5 = 9
9 + 6y + -5y2 = -5 + 5y2 + 5 + -5y2

Reorder the terms:
9 + 6y + -5y2 = -5 + 5 + 5y2 + -5y2

Combine like terms: -5 + 5 = 0
9 + 6y + -5y2 = 0 + 5y2 + -5y2
9 + 6y + -5y2 = 5y2 + -5y2

Combine like terms: 5y2 + -5y2 = 0
9 + 6y + -5y2 = 0

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

Divide each side by '-5'.
-1.8 + -1.2y + y2 = 0

Move the constant term to the right:

Add '1.8' to each side of the equation.
-1.8 + -1.2y + 1.8 + y2 = 0 + 1.8

Reorder the terms:
-1.8 + 1.8 + -1.2y + y2 = 0 + 1.8

Combine like terms: -1.8 + 1.8 = 0.0
0.0 + -1.2y + y2 = 0 + 1.8
-1.2y + y2 = 0 + 1.8

Combine like terms: 0 + 1.8 = 1.8
-1.2y + y2 = 1.8

The y term is -1.2y.  Take half its coefficient (-0.6).
Square it (0.36) and add it to both sides.

Add '0.36' to each side of the equation.
-1.2y + 0.36 + y2 = 1.8 + 0.36

Reorder the terms:
0.36 + -1.2y + y2 = 1.8 + 0.36

Combine like terms: 1.8 + 0.36 = 2.16
0.36 + -1.2y + y2 = 2.16

Factor a perfect square on the left side:
(y + -0.6)(y + -0.6) = 2.16

Calculate the square root of the right side: 1.469693846

Break this problem into two subproblems by setting 
(y + -0.6) equal to 1.469693846 and -1.469693846.

Subproblem 1
y + -0.6 = 1.469693846

Simplifying
y + -0.6 = 1.469693846

Reorder the terms:
-0.6 + y = 1.469693846

Solving
-0.6 + y = 1.469693846

Solving for variable 'y'.

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

Add '0.6' to each side of the equation.
-0.6 + 0.6 + y = 1.469693846 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + y = 1.469693846 + 0.6
y = 1.469693846 + 0.6

Combine like terms: 1.469693846 + 0.6 = 2.069693846
y = 2.069693846

Simplifying
y = 2.069693846

Subproblem 2
y + -0.6 = -1.469693846

Simplifying
y + -0.6 = -1.469693846

Reorder the terms:
-0.6 + y = -1.469693846

Solving
-0.6 + y = -1.469693846

Solving for variable 'y'.

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

Add '0.6' to each side of the equation.
-0.6 + 0.6 + y = -1.469693846 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + y = -1.469693846 + 0.6
y = -1.469693846 + 0.6

Combine like terms: -1.469693846 + 0.6 = -0.869693846
y = -0.869693846

Simplifying
y = -0.869693846

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.069693846, -0.869693846}