12y^2=40y-3

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Solution for 12y^2=40y-3 equation:


Simplifying
12y2 = 40y + -3

Reorder the terms:
12y2 = -3 + 40y

Solving
12y2 = -3 + 40y

Solving for variable 'y'.

Reorder the terms:
3 + -40y + 12y2 = -3 + 40y + 3 + -40y

Reorder the terms:
3 + -40y + 12y2 = -3 + 3 + 40y + -40y

Combine like terms: -3 + 3 = 0
3 + -40y + 12y2 = 0 + 40y + -40y
3 + -40y + 12y2 = 40y + -40y

Combine like terms: 40y + -40y = 0
3 + -40y + 12y2 = 0

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

Divide each side by '12'.
0.25 + -3.333333333y + y2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -3.333333333y + -0.25 + y2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -3.333333333y + y2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -3.333333333y + y2 = 0 + -0.25
-3.333333333y + y2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-3.333333333y + y2 = -0.25

The y term is -3.333333333y.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333y + 2.777777779 + y2 = -0.25 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333y + y2 = -0.25 + 2.777777779

Combine like terms: -0.25 + 2.777777779 = 2.527777779
2.777777779 + -3.333333333y + y2 = 2.527777779

Factor a perfect square on the left side:
(y + -1.666666667)(y + -1.666666667) = 2.527777779

Calculate the square root of the right side: 1.589898669

Break this problem into two subproblems by setting 
(y + -1.666666667) equal to 1.589898669 and -1.589898669.

Subproblem 1

y + -1.666666667 = 1.589898669 Simplifying y + -1.666666667 = 1.589898669 Reorder the terms: -1.666666667 + y = 1.589898669 Solving -1.666666667 + y = 1.589898669 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + y = 1.589898669 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = 1.589898669 + 1.666666667 y = 1.589898669 + 1.666666667 Combine like terms: 1.589898669 + 1.666666667 = 3.256565336 y = 3.256565336 Simplifying y = 3.256565336

Subproblem 2

y + -1.666666667 = -1.589898669 Simplifying y + -1.666666667 = -1.589898669 Reorder the terms: -1.666666667 + y = -1.589898669 Solving -1.666666667 + y = -1.589898669 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + y = -1.589898669 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = -1.589898669 + 1.666666667 y = -1.589898669 + 1.666666667 Combine like terms: -1.589898669 + 1.666666667 = 0.076767998 y = 0.076767998 Simplifying y = 0.076767998

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.256565336, 0.076767998}

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