6y^2-2y=9

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


Simplifying
6y2 + -2y = 9

Reorder the terms:
-2y + 6y2 = 9

Solving
-2y + 6y2 = 9

Solving for variable 'y'.

Reorder the terms:
-9 + -2y + 6y2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + -2y + 6y2 = 0

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

Divide each side by '6'.
-1.5 + -0.3333333333y + y2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.3333333333y + 1.5 + y2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.3333333333y + y2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.3333333333y + y2 = 0 + 1.5
-0.3333333333y + y2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-0.3333333333y + y2 = 1.5

The y term is -0.3333333333y.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333y + 0.02777777779 + y2 = 1.5 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333y + y2 = 1.5 + 0.02777777779

Combine like terms: 1.5 + 0.02777777779 = 1.52777777779
0.02777777779 + -0.3333333333y + y2 = 1.52777777779

Factor a perfect square on the left side:
(y + -0.1666666667)(y + -0.1666666667) = 1.52777777779

Calculate the square root of the right side: 1.236033081

Break this problem into two subproblems by setting 
(y + -0.1666666667) equal to 1.236033081 and -1.236033081.

Subproblem 1

y + -0.1666666667 = 1.236033081 Simplifying y + -0.1666666667 = 1.236033081 Reorder the terms: -0.1666666667 + y = 1.236033081 Solving -0.1666666667 + y = 1.236033081 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + y = 1.236033081 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = 1.236033081 + 0.1666666667 y = 1.236033081 + 0.1666666667 Combine like terms: 1.236033081 + 0.1666666667 = 1.4026997477 y = 1.4026997477 Simplifying y = 1.4026997477

Subproblem 2

y + -0.1666666667 = -1.236033081 Simplifying y + -0.1666666667 = -1.236033081 Reorder the terms: -0.1666666667 + y = -1.236033081 Solving -0.1666666667 + y = -1.236033081 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + y = -1.236033081 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = -1.236033081 + 0.1666666667 y = -1.236033081 + 0.1666666667 Combine like terms: -1.236033081 + 0.1666666667 = -1.0693664143 y = -1.0693664143 Simplifying y = -1.0693664143

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.4026997477, -1.0693664143}

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