(6y^2-7y-5)+(2y^2-4y+3)=0

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Solution for (6y^2-7y-5)+(2y^2-4y+3)=0 equation:


Simplifying
(6y2 + -7y + -5) + (2y2 + -4y + 3) = 0

Reorder the terms:
(-5 + -7y + 6y2) + (2y2 + -4y + 3) = 0

Remove parenthesis around (-5 + -7y + 6y2)
-5 + -7y + 6y2 + (2y2 + -4y + 3) = 0

Reorder the terms:
-5 + -7y + 6y2 + (3 + -4y + 2y2) = 0

Remove parenthesis around (3 + -4y + 2y2)
-5 + -7y + 6y2 + 3 + -4y + 2y2 = 0

Reorder the terms:
-5 + 3 + -7y + -4y + 6y2 + 2y2 = 0

Combine like terms: -5 + 3 = -2
-2 + -7y + -4y + 6y2 + 2y2 = 0

Combine like terms: -7y + -4y = -11y
-2 + -11y + 6y2 + 2y2 = 0

Combine like terms: 6y2 + 2y2 = 8y2
-2 + -11y + 8y2 = 0

Solving
-2 + -11y + 8y2 = 0

Solving for variable 'y'.

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

Divide each side by '8'.
-0.25 + -1.375y + y2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1.375y + 0.25 + y2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1.375y + y2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1.375y + y2 = 0 + 0.25
-1.375y + y2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1.375y + y2 = 0.25

The y term is -1.375y.  Take half its coefficient (-0.6875).
Square it (0.47265625) and add it to both sides.

Add '0.47265625' to each side of the equation.
-1.375y + 0.47265625 + y2 = 0.25 + 0.47265625

Reorder the terms:
0.47265625 + -1.375y + y2 = 0.25 + 0.47265625

Combine like terms: 0.25 + 0.47265625 = 0.72265625
0.47265625 + -1.375y + y2 = 0.72265625

Factor a perfect square on the left side:
(y + -0.6875)(y + -0.6875) = 0.72265625

Calculate the square root of the right side: 0.850091907

Break this problem into two subproblems by setting 
(y + -0.6875) equal to 0.850091907 and -0.850091907.

Subproblem 1

y + -0.6875 = 0.850091907 Simplifying y + -0.6875 = 0.850091907 Reorder the terms: -0.6875 + y = 0.850091907 Solving -0.6875 + y = 0.850091907 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + y = 0.850091907 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + y = 0.850091907 + 0.6875 y = 0.850091907 + 0.6875 Combine like terms: 0.850091907 + 0.6875 = 1.537591907 y = 1.537591907 Simplifying y = 1.537591907

Subproblem 2

y + -0.6875 = -0.850091907 Simplifying y + -0.6875 = -0.850091907 Reorder the terms: -0.6875 + y = -0.850091907 Solving -0.6875 + y = -0.850091907 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + y = -0.850091907 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + y = -0.850091907 + 0.6875 y = -0.850091907 + 0.6875 Combine like terms: -0.850091907 + 0.6875 = -0.162591907 y = -0.162591907 Simplifying y = -0.162591907

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.537591907, -0.162591907}

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