3Y^2-3Y-2=0

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Solution for 3Y^2-3Y-2=0 equation:


Simplifying
3Y2 + -3Y + -2 = 0

Reorder the terms:
-2 + -3Y + 3Y2 = 0

Solving
-2 + -3Y + 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 + -1Y + Y2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -1Y + 0.6666666667 + Y2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -1Y + Y2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -1Y + Y2 = 0 + 0.6666666667
-1Y + Y2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-1Y + Y2 = 0.6666666667

The Y term is -1Y.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1Y + 0.25 + Y2 = 0.6666666667 + 0.25

Reorder the terms:
0.25 + -1Y + Y2 = 0.6666666667 + 0.25

Combine like terms: 0.6666666667 + 0.25 = 0.9166666667
0.25 + -1Y + Y2 = 0.9166666667

Factor a perfect square on the left side:
(Y + -0.5)(Y + -0.5) = 0.9166666667

Calculate the square root of the right side: 0.957427108

Break this problem into two subproblems by setting 
(Y + -0.5) equal to 0.957427108 and -0.957427108.

Subproblem 1

Y + -0.5 = 0.957427108 Simplifying Y + -0.5 = 0.957427108 Reorder the terms: -0.5 + Y = 0.957427108 Solving -0.5 + Y = 0.957427108 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + Y = 0.957427108 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + Y = 0.957427108 + 0.5 Y = 0.957427108 + 0.5 Combine like terms: 0.957427108 + 0.5 = 1.457427108 Y = 1.457427108 Simplifying Y = 1.457427108

Subproblem 2

Y + -0.5 = -0.957427108 Simplifying Y + -0.5 = -0.957427108 Reorder the terms: -0.5 + Y = -0.957427108 Solving -0.5 + Y = -0.957427108 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + Y = -0.957427108 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + Y = -0.957427108 + 0.5 Y = -0.957427108 + 0.5 Combine like terms: -0.957427108 + 0.5 = -0.457427108 Y = -0.457427108 Simplifying Y = -0.457427108

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {1.457427108, -0.457427108}

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