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

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


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

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

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

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

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

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

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

Combine like terms: 3y2 + -2y2 = 1y2
-6 + 3y + 1y2 = 0

Solving
-6 + 3y + 1y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 3y + 6 + y2 = 0 + 6

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

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

Combine like terms: 0 + 6 = 6
3y + y2 = 6

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 6 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 6 + 2.25

Combine like terms: 6 + 2.25 = 8.25
2.25 + 3y + y2 = 8.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(y + 1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

y + 1.5 = 2.872281323 Simplifying y + 1.5 = 2.872281323 Reorder the terms: 1.5 + y = 2.872281323 Solving 1.5 + y = 2.872281323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 2.872281323 + -1.5 y = 2.872281323 + -1.5 Combine like terms: 2.872281323 + -1.5 = 1.372281323 y = 1.372281323 Simplifying y = 1.372281323

Subproblem 2

y + 1.5 = -2.872281323 Simplifying y + 1.5 = -2.872281323 Reorder the terms: 1.5 + y = -2.872281323 Solving 1.5 + y = -2.872281323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -2.872281323 + -1.5 y = -2.872281323 + -1.5 Combine like terms: -2.872281323 + -1.5 = -4.372281323 y = -4.372281323 Simplifying y = -4.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.372281323, -4.372281323}

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