15-y^2-10y-8+8y=0

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Solution for 15-y^2-10y-8+8y=0 equation: 


Simplifying
15 + -1y2 + -10y + -8 + 8y = 0

Reorder the terms:
15 + -8 + -10y + 8y + -1y2 = 0

Combine like terms: 15 + -8 = 7
7 + -10y + 8y + -1y2 = 0

Combine like terms: -10y + 8y = -2y
7 + -2y + -1y2 = 0

Solving
7 + -2y + -1y2 = 0

Solving for variable 'y'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-7 + 2y + y2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 2y + 7 + y2 = 0 + 7

Reorder the terms:
-7 + 7 + 2y + y2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 2y + y2 = 0 + 7
2y + y2 = 0 + 7

Combine like terms: 0 + 7 = 7
2y + y2 = 7

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

Add '1' to each side of the equation.
2y + 1 + y2 = 7 + 1

Reorder the terms:
1 + 2y + y2 = 7 + 1

Combine like terms: 7 + 1 = 8
1 + 2y + y2 = 8

Factor a perfect square on the left side:
(y + 1)(y + 1) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(y + 1) equal to 2.828427125 and -2.828427125.


Subproblem 1
y + 1 = 2.828427125

Simplifying
y + 1 = 2.828427125

Reorder the terms:
1 + y = 2.828427125

Solving
1 + y = 2.828427125

Solving for variable 'y'.

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

Add '-1' to each side of the equation.
1 + -1 + y = 2.828427125 + -1

Combine like terms: 1 + -1 = 0
0 + y = 2.828427125 + -1
y = 2.828427125 + -1

Combine like terms: 2.828427125 + -1 = 1.828427125
y = 1.828427125

Simplifying
y = 1.828427125


Subproblem 2
y + 1 = -2.828427125

Simplifying
y + 1 = -2.828427125

Reorder the terms:
1 + y = -2.828427125

Solving
1 + y = -2.828427125

Solving for variable 'y'.

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

Add '-1' to each side of the equation.
1 + -1 + y = -2.828427125 + -1

Combine like terms: 1 + -1 = 0
0 + y = -2.828427125 + -1
y = -2.828427125 + -1

Combine like terms: -2.828427125 + -1 = -3.828427125
y = -3.828427125

Simplifying
y = -3.828427125


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.828427125, -3.828427125}

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