(3y^2-5y+2)-(5y^2-2y-2)=0

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


Simplifying
(3y2 + -5y + 2) + -1(5y2 + -2y + -2) = 0

Reorder the terms:
(2 + -5y + 3y2) + -1(5y2 + -2y + -2) = 0

Remove parenthesis around (2 + -5y + 3y2)
2 + -5y + 3y2 + -1(5y2 + -2y + -2) = 0

Reorder the terms:
2 + -5y + 3y2 + -1(-2 + -2y + 5y2) = 0
2 + -5y + 3y2 + (-2 * -1 + -2y * -1 + 5y2 * -1) = 0
2 + -5y + 3y2 + (2 + 2y + -5y2) = 0

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

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

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

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

Solving
4 + -3y + -2y2 = 0

Solving for variable 'y'.

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

Divide each side by '-2'.
-2 + 1.5y + y2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 1.5y + 2 + y2 = 0 + 2

Reorder the terms:
-2 + 2 + 1.5y + y2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 1.5y + y2 = 0 + 2
1.5y + y2 = 0 + 2

Combine like terms: 0 + 2 = 2
1.5y + y2 = 2

The y term is 1.5y.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5y + 0.5625 + y2 = 2 + 0.5625

Reorder the terms:
0.5625 + 1.5y + y2 = 2 + 0.5625

Combine like terms: 2 + 0.5625 = 2.5625
0.5625 + 1.5y + y2 = 2.5625

Factor a perfect square on the left side:
(y + 0.75)(y + 0.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(y + 0.75) equal to 1.600781059 and -1.600781059.

Subproblem 1

y + 0.75 = 1.600781059 Simplifying y + 0.75 = 1.600781059 Reorder the terms: 0.75 + y = 1.600781059 Solving 0.75 + y = 1.600781059 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 1.600781059 + -0.75 y = 1.600781059 + -0.75 Combine like terms: 1.600781059 + -0.75 = 0.850781059 y = 0.850781059 Simplifying y = 0.850781059

Subproblem 2

y + 0.75 = -1.600781059 Simplifying y + 0.75 = -1.600781059 Reorder the terms: 0.75 + y = -1.600781059 Solving 0.75 + y = -1.600781059 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -1.600781059 + -0.75 y = -1.600781059 + -0.75 Combine like terms: -1.600781059 + -0.75 = -2.350781059 y = -2.350781059 Simplifying y = -2.350781059

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.850781059, -2.350781059}

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