2y^2-8y+15-y^2+36y-12=0

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Solution for 2y^2-8y+15-y^2+36y-12=0 equation:


Simplifying
2y2 + -8y + 15 + -1y2 + 36y + -12 = 0

Reorder the terms:
15 + -12 + -8y + 36y + 2y2 + -1y2 = 0

Combine like terms: 15 + -12 = 3
3 + -8y + 36y + 2y2 + -1y2 = 0

Combine like terms: -8y + 36y = 28y
3 + 28y + 2y2 + -1y2 = 0

Combine like terms: 2y2 + -1y2 = 1y2
3 + 28y + 1y2 = 0

Solving
3 + 28y + 1y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 28y + -3 + y2 = 0 + -3

Reorder the terms:
3 + -3 + 28y + y2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 28y + y2 = 0 + -3
28y + y2 = 0 + -3

Combine like terms: 0 + -3 = -3
28y + y2 = -3

The y term is 28y.  Take half its coefficient (14).
Square it (196) and add it to both sides.

Add '196' to each side of the equation.
28y + 196 + y2 = -3 + 196

Reorder the terms:
196 + 28y + y2 = -3 + 196

Combine like terms: -3 + 196 = 193
196 + 28y + y2 = 193

Factor a perfect square on the left side:
(y + 14)(y + 14) = 193

Calculate the square root of the right side: 13.892443989

Break this problem into two subproblems by setting 
(y + 14) equal to 13.892443989 and -13.892443989.

Subproblem 1

y + 14 = 13.892443989 Simplifying y + 14 = 13.892443989 Reorder the terms: 14 + y = 13.892443989 Solving 14 + y = 13.892443989 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + y = 13.892443989 + -14 Combine like terms: 14 + -14 = 0 0 + y = 13.892443989 + -14 y = 13.892443989 + -14 Combine like terms: 13.892443989 + -14 = -0.107556011 y = -0.107556011 Simplifying y = -0.107556011

Subproblem 2

y + 14 = -13.892443989 Simplifying y + 14 = -13.892443989 Reorder the terms: 14 + y = -13.892443989 Solving 14 + y = -13.892443989 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + y = -13.892443989 + -14 Combine like terms: 14 + -14 = 0 0 + y = -13.892443989 + -14 y = -13.892443989 + -14 Combine like terms: -13.892443989 + -14 = -27.892443989 y = -27.892443989 Simplifying y = -27.892443989

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.107556011, -27.892443989}

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