3y^2-4y-6=0

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Solution for 3y^2-4y-6=0 equation:


Simplifying
3y2 + -4y + -6 = 0

Reorder the terms:
-6 + -4y + 3y2 = 0

Solving
-6 + -4y + 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'.
-2 + -1.333333333y + y2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.333333333y + 2 + y2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.333333333y + y2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.333333333y + y2 = 0 + 2
-1.333333333y + y2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.333333333y + y2 = 2

The y term is -1.333333333y.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333y + 0.4444444442 + y2 = 2 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333y + y2 = 2 + 0.4444444442

Combine like terms: 2 + 0.4444444442 = 2.4444444442
0.4444444442 + -1.333333333y + y2 = 2.4444444442

Factor a perfect square on the left side:
(y + -0.6666666665)(y + -0.6666666665) = 2.4444444442

Calculate the square root of the right side: 1.56347192

Break this problem into two subproblems by setting 
(y + -0.6666666665) equal to 1.56347192 and -1.56347192.

Subproblem 1

y + -0.6666666665 = 1.56347192 Simplifying y + -0.6666666665 = 1.56347192 Reorder the terms: -0.6666666665 + y = 1.56347192 Solving -0.6666666665 + y = 1.56347192 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = 1.56347192 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = 1.56347192 + 0.6666666665 y = 1.56347192 + 0.6666666665 Combine like terms: 1.56347192 + 0.6666666665 = 2.2301385865 y = 2.2301385865 Simplifying y = 2.2301385865

Subproblem 2

y + -0.6666666665 = -1.56347192 Simplifying y + -0.6666666665 = -1.56347192 Reorder the terms: -0.6666666665 + y = -1.56347192 Solving -0.6666666665 + y = -1.56347192 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = -1.56347192 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = -1.56347192 + 0.6666666665 y = -1.56347192 + 0.6666666665 Combine like terms: -1.56347192 + 0.6666666665 = -0.8968052535 y = -0.8968052535 Simplifying y = -0.8968052535

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.2301385865, -0.8968052535}

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