7-y^2+2y+8y^2=-5y+6+5y^2

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Solution for 7-y^2+2y+8y^2=-5y+6+5y^2 equation:


Simplifying
7 + -1y2 + 2y + 8y2 = -5y + 6 + 5y2

Reorder the terms:
7 + 2y + -1y2 + 8y2 = -5y + 6 + 5y2

Combine like terms: -1y2 + 8y2 = 7y2
7 + 2y + 7y2 = -5y + 6 + 5y2

Reorder the terms:
7 + 2y + 7y2 = 6 + -5y + 5y2

Solving
7 + 2y + 7y2 = 6 + -5y + 5y2

Solving for variable 'y'.

Reorder the terms:
7 + -6 + 2y + 5y + 7y2 + -5y2 = 6 + -5y + 5y2 + -6 + 5y + -5y2

Combine like terms: 7 + -6 = 1
1 + 2y + 5y + 7y2 + -5y2 = 6 + -5y + 5y2 + -6 + 5y + -5y2

Combine like terms: 2y + 5y = 7y
1 + 7y + 7y2 + -5y2 = 6 + -5y + 5y2 + -6 + 5y + -5y2

Combine like terms: 7y2 + -5y2 = 2y2
1 + 7y + 2y2 = 6 + -5y + 5y2 + -6 + 5y + -5y2

Reorder the terms:
1 + 7y + 2y2 = 6 + -6 + -5y + 5y + 5y2 + -5y2

Combine like terms: 6 + -6 = 0
1 + 7y + 2y2 = 0 + -5y + 5y + 5y2 + -5y2
1 + 7y + 2y2 = -5y + 5y + 5y2 + -5y2

Combine like terms: -5y + 5y = 0
1 + 7y + 2y2 = 0 + 5y2 + -5y2
1 + 7y + 2y2 = 5y2 + -5y2

Combine like terms: 5y2 + -5y2 = 0
1 + 7y + 2y2 = 0

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

Divide each side by '2'.
0.5 + 3.5y + y2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 3.5y + -0.5 + y2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 3.5y + y2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 3.5y + y2 = 0 + -0.5
3.5y + y2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
3.5y + y2 = -0.5

The y term is 3.5y.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5y + 3.0625 + y2 = -0.5 + 3.0625

Reorder the terms:
3.0625 + 3.5y + y2 = -0.5 + 3.0625

Combine like terms: -0.5 + 3.0625 = 2.5625
3.0625 + 3.5y + y2 = 2.5625

Factor a perfect square on the left side:
(y + 1.75)(y + 1.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(y + 1.75) equal to 1.600781059 and -1.600781059.

Subproblem 1

y + 1.75 = 1.600781059 Simplifying y + 1.75 = 1.600781059 Reorder the terms: 1.75 + y = 1.600781059 Solving 1.75 + y = 1.600781059 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = 1.600781059 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = 1.600781059 + -1.75 y = 1.600781059 + -1.75 Combine like terms: 1.600781059 + -1.75 = -0.149218941 y = -0.149218941 Simplifying y = -0.149218941

Subproblem 2

y + 1.75 = -1.600781059 Simplifying y + 1.75 = -1.600781059 Reorder the terms: 1.75 + y = -1.600781059 Solving 1.75 + y = -1.600781059 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = -1.600781059 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = -1.600781059 + -1.75 y = -1.600781059 + -1.75 Combine like terms: -1.600781059 + -1.75 = -3.350781059 y = -3.350781059 Simplifying y = -3.350781059

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.149218941, -3.350781059}

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