6y^2-4y=9

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Solution for 6y^2-4y=9 equation:


Simplifying
6y2 + -4y = 9

Reorder the terms:
-4y + 6y2 = 9

Solving
-4y + 6y2 = 9

Solving for variable 'y'.

Reorder the terms:
-9 + -4y + 6y2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + -4y + 6y2 = 0

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

Divide each side by '6'.
-1.5 + -0.6666666667y + y2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.6666666667y + 1.5 + y2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.6666666667y + y2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.6666666667y + y2 = 0 + 1.5
-0.6666666667y + y2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-0.6666666667y + y2 = 1.5

The y term is -0.6666666667y.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667y + 0.1111111112 + y2 = 1.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667y + y2 = 1.5 + 0.1111111112

Combine like terms: 1.5 + 0.1111111112 = 1.6111111112
0.1111111112 + -0.6666666667y + y2 = 1.6111111112

Factor a perfect square on the left side:
(y + -0.3333333334)(y + -0.3333333334) = 1.6111111112

Calculate the square root of the right side: 1.269295518

Break this problem into two subproblems by setting 
(y + -0.3333333334) equal to 1.269295518 and -1.269295518.

Subproblem 1

y + -0.3333333334 = 1.269295518 Simplifying y + -0.3333333334 = 1.269295518 Reorder the terms: -0.3333333334 + y = 1.269295518 Solving -0.3333333334 + y = 1.269295518 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = 1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = 1.269295518 + 0.3333333334 y = 1.269295518 + 0.3333333334 Combine like terms: 1.269295518 + 0.3333333334 = 1.6026288514 y = 1.6026288514 Simplifying y = 1.6026288514

Subproblem 2

y + -0.3333333334 = -1.269295518 Simplifying y + -0.3333333334 = -1.269295518 Reorder the terms: -0.3333333334 + y = -1.269295518 Solving -0.3333333334 + y = -1.269295518 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = -1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = -1.269295518 + 0.3333333334 y = -1.269295518 + 0.3333333334 Combine like terms: -1.269295518 + 0.3333333334 = -0.9359621846 y = -0.9359621846 Simplifying y = -0.9359621846

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.6026288514, -0.9359621846}

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