(-6y^2+5y+8)-(-3y^2+1)=0

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Solution for (-6y^2+5y+8)-(-3y^2+1)=0 equation:


Simplifying
(-6y2 + 5y + 8) + -1(-3y2 + 1) = 0

Reorder the terms:
(8 + 5y + -6y2) + -1(-3y2 + 1) = 0

Remove parenthesis around (8 + 5y + -6y2)
8 + 5y + -6y2 + -1(-3y2 + 1) = 0

Reorder the terms:
8 + 5y + -6y2 + -1(1 + -3y2) = 0
8 + 5y + -6y2 + (1 * -1 + -3y2 * -1) = 0
8 + 5y + -6y2 + (-1 + 3y2) = 0

Reorder the terms:
8 + -1 + 5y + -6y2 + 3y2 = 0

Combine like terms: 8 + -1 = 7
7 + 5y + -6y2 + 3y2 = 0

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

Solving
7 + 5y + -3y2 = 0

Solving for variable 'y'.

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-2.333333333 + -1.666666667y + y2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -1.666666667y + 2.333333333 + y2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -1.666666667y + y2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -1.666666667y + y2 = 0 + 2.333333333
-1.666666667y + y2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-1.666666667y + y2 = 2.333333333

The y term is -1.666666667y.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667y + 0.6944444447 + y2 = 2.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667y + y2 = 2.333333333 + 0.6944444447

Combine like terms: 2.333333333 + 0.6944444447 = 3.0277777777
0.6944444447 + -1.666666667y + y2 = 3.0277777777

Factor a perfect square on the left side:
(y + -0.8333333335)(y + -0.8333333335) = 3.0277777777

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(y + -0.8333333335) equal to 1.740051085 and -1.740051085.

Subproblem 1

y + -0.8333333335 = 1.740051085 Simplifying y + -0.8333333335 = 1.740051085 Reorder the terms: -0.8333333335 + y = 1.740051085 Solving -0.8333333335 + y = 1.740051085 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = 1.740051085 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = 1.740051085 + 0.8333333335 y = 1.740051085 + 0.8333333335 Combine like terms: 1.740051085 + 0.8333333335 = 2.5733844185 y = 2.5733844185 Simplifying y = 2.5733844185

Subproblem 2

y + -0.8333333335 = -1.740051085 Simplifying y + -0.8333333335 = -1.740051085 Reorder the terms: -0.8333333335 + y = -1.740051085 Solving -0.8333333335 + y = -1.740051085 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = -1.740051085 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = -1.740051085 + 0.8333333335 y = -1.740051085 + 0.8333333335 Combine like terms: -1.740051085 + 0.8333333335 = -0.9067177515 y = -0.9067177515 Simplifying y = -0.9067177515

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.5733844185, -0.9067177515}

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