3y-6-3y^2+5y+3=

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Solution for 3y-6-3y^2+5y+3= equation:


Simplifying
3y + -6 + -3y2 + 5y + 3 = 0

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

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

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

Solving
-3 + 8y + -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'.
1 + -2.666666667y + y2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -2.666666667y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + -2.666666667y + y2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2.666666667y + y2 = 0 + -1
-2.666666667y + y2 = 0 + -1

Combine like terms: 0 + -1 = -1
-2.666666667y + y2 = -1

The y term is -2.666666667y.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667y + 1.777777780 + y2 = -1 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667y + y2 = -1 + 1.777777780

Combine like terms: -1 + 1.777777780 = 0.77777778
1.777777780 + -2.666666667y + y2 = 0.77777778

Factor a perfect square on the left side:
(y + -1.333333334)(y + -1.333333334) = 0.77777778

Calculate the square root of the right side: 0.881917105

Break this problem into two subproblems by setting 
(y + -1.333333334) equal to 0.881917105 and -0.881917105.

Subproblem 1

y + -1.333333334 = 0.881917105 Simplifying y + -1.333333334 = 0.881917105 Reorder the terms: -1.333333334 + y = 0.881917105 Solving -1.333333334 + y = 0.881917105 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = 0.881917105 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 0.881917105 + 1.333333334 y = 0.881917105 + 1.333333334 Combine like terms: 0.881917105 + 1.333333334 = 2.215250439 y = 2.215250439 Simplifying y = 2.215250439

Subproblem 2

y + -1.333333334 = -0.881917105 Simplifying y + -1.333333334 = -0.881917105 Reorder the terms: -1.333333334 + y = -0.881917105 Solving -1.333333334 + y = -0.881917105 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = -0.881917105 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -0.881917105 + 1.333333334 y = -0.881917105 + 1.333333334 Combine like terms: -0.881917105 + 1.333333334 = 0.451416229 y = 0.451416229 Simplifying y = 0.451416229

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.215250439, 0.451416229}

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