3y^2-8y=12

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Solution for 3y^2-8y=12 equation:


Simplifying
3y2 + -8y = 12

Reorder the terms:
-8y + 3y2 = 12

Solving
-8y + 3y2 = 12

Solving for variable 'y'.

Reorder the terms:
-12 + -8y + 3y2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + -8y + 3y2 = 0

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

Divide each side by '3'.
-4 + -2.666666667y + y2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -2.666666667y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + -2.666666667y + y2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -2.666666667y + y2 = 0 + 4
-2.666666667y + y2 = 0 + 4

Combine like terms: 0 + 4 = 4
-2.666666667y + y2 = 4

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 = 4 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667y + y2 = 4 + 1.777777780

Combine like terms: 4 + 1.777777780 = 5.77777778
1.777777780 + -2.666666667y + y2 = 5.77777778

Factor a perfect square on the left side:
(y + -1.333333334)(y + -1.333333334) = 5.77777778

Calculate the square root of the right side: 2.403700851

Break this problem into two subproblems by setting 
(y + -1.333333334) equal to 2.403700851 and -2.403700851.

Subproblem 1

y + -1.333333334 = 2.403700851 Simplifying y + -1.333333334 = 2.403700851 Reorder the terms: -1.333333334 + y = 2.403700851 Solving -1.333333334 + y = 2.403700851 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 = 2.403700851 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 2.403700851 + 1.333333334 y = 2.403700851 + 1.333333334 Combine like terms: 2.403700851 + 1.333333334 = 3.737034185 y = 3.737034185 Simplifying y = 3.737034185

Subproblem 2

y + -1.333333334 = -2.403700851 Simplifying y + -1.333333334 = -2.403700851 Reorder the terms: -1.333333334 + y = -2.403700851 Solving -1.333333334 + y = -2.403700851 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 = -2.403700851 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -2.403700851 + 1.333333334 y = -2.403700851 + 1.333333334 Combine like terms: -2.403700851 + 1.333333334 = -1.070367517 y = -1.070367517 Simplifying y = -1.070367517

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.737034185, -1.070367517}

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