3y^2-9y-2=0

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Solution for 3y^2-9y-2=0 equation:


Simplifying
3y2 + -9y + -2 = 0

Reorder the terms:
-2 + -9y + 3y2 = 0

Solving
-2 + -9y + 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'.
-0.6666666667 + -3y + y2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -3y + 0.6666666667 + y2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -3y + y2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -3y + y2 = 0 + 0.6666666667
-3y + y2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-3y + y2 = 0.6666666667

The y term is -3y.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3y + 2.25 + y2 = 0.6666666667 + 2.25

Reorder the terms:
2.25 + -3y + y2 = 0.6666666667 + 2.25

Combine like terms: 0.6666666667 + 2.25 = 2.9166666667
2.25 + -3y + y2 = 2.9166666667

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 2.9166666667

Calculate the square root of the right side: 1.707825128

Break this problem into two subproblems by setting 
(y + -1.5) equal to 1.707825128 and -1.707825128.

Subproblem 1

y + -1.5 = 1.707825128 Simplifying y + -1.5 = 1.707825128 Reorder the terms: -1.5 + y = 1.707825128 Solving -1.5 + y = 1.707825128 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = 1.707825128 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 1.707825128 + 1.5 y = 1.707825128 + 1.5 Combine like terms: 1.707825128 + 1.5 = 3.207825128 y = 3.207825128 Simplifying y = 3.207825128

Subproblem 2

y + -1.5 = -1.707825128 Simplifying y + -1.5 = -1.707825128 Reorder the terms: -1.5 + y = -1.707825128 Solving -1.5 + y = -1.707825128 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = -1.707825128 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -1.707825128 + 1.5 y = -1.707825128 + 1.5 Combine like terms: -1.707825128 + 1.5 = -0.207825128 y = -0.207825128 Simplifying y = -0.207825128

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.207825128, -0.207825128}

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