6y^2-11y-3=0

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Solution for 6y^2-11y-3=0 equation:


Simplifying
6y2 + -11y + -3 = 0

Reorder the terms:
-3 + -11y + 6y2 = 0

Solving
-3 + -11y + 6y2 = 0

Solving for variable 'y'.

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

Divide each side by '6'.
-0.5 + -1.833333333y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1.833333333y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1.833333333y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1.833333333y + y2 = 0 + 0.5
-1.833333333y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1.833333333y + y2 = 0.5

The y term is -1.833333333y.  Take half its coefficient (-0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
-1.833333333y + 0.8402777775 + y2 = 0.5 + 0.8402777775

Reorder the terms:
0.8402777775 + -1.833333333y + y2 = 0.5 + 0.8402777775

Combine like terms: 0.5 + 0.8402777775 = 1.3402777775
0.8402777775 + -1.833333333y + y2 = 1.3402777775

Factor a perfect square on the left side:
(y + -0.9166666665)(y + -0.9166666665) = 1.3402777775

Calculate the square root of the right side: 1.157703666

Break this problem into two subproblems by setting 
(y + -0.9166666665) equal to 1.157703666 and -1.157703666.

Subproblem 1

y + -0.9166666665 = 1.157703666 Simplifying y + -0.9166666665 = 1.157703666 Reorder the terms: -0.9166666665 + y = 1.157703666 Solving -0.9166666665 + y = 1.157703666 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + y = 1.157703666 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + y = 1.157703666 + 0.9166666665 y = 1.157703666 + 0.9166666665 Combine like terms: 1.157703666 + 0.9166666665 = 2.0743703325 y = 2.0743703325 Simplifying y = 2.0743703325

Subproblem 2

y + -0.9166666665 = -1.157703666 Simplifying y + -0.9166666665 = -1.157703666 Reorder the terms: -0.9166666665 + y = -1.157703666 Solving -0.9166666665 + y = -1.157703666 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + y = -1.157703666 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + y = -1.157703666 + 0.9166666665 y = -1.157703666 + 0.9166666665 Combine like terms: -1.157703666 + 0.9166666665 = -0.2410369995 y = -0.2410369995 Simplifying y = -0.2410369995

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.0743703325, -0.2410369995}

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