3=6y-2y^2

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Solution for 3=6y-2y^2 equation:


Simplifying
3 = 6y + -2y2

Solving
3 = 6y + -2y2

Solving for variable 'y'.

Reorder the terms:
3 + -6y + 2y2 = 6y + -6y + -2y2 + 2y2

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

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

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

Divide each side by '2'.
1.5 + -3y + y2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3y + -1.5 + y2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3y + y2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3y + y2 = 0 + -1.5
-3y + y2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3y + y2 = -1.5

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 = -1.5 + 2.25

Reorder the terms:
2.25 + -3y + y2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + -3y + y2 = 0.75

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(y + -1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

y + -1.5 = 0.866025404 Simplifying y + -1.5 = 0.866025404 Reorder the terms: -1.5 + y = 0.866025404 Solving -1.5 + y = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 0.866025404 + 1.5 y = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 y = 2.366025404 Simplifying y = 2.366025404

Subproblem 2

y + -1.5 = -0.866025404 Simplifying y + -1.5 = -0.866025404 Reorder the terms: -1.5 + y = -0.866025404 Solving -1.5 + y = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -0.866025404 + 1.5 y = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 y = 0.633974596 Simplifying y = 0.633974596

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.366025404, 0.633974596}

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