6y^2-4y-3=0

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


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

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

Solving
-3 + -4y + 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 + -0.6666666667y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.6666666667y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.6666666667y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.6666666667y + y2 = 0 + 0.5
-0.6666666667y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.6666666667y + y2 = 0.5

The y term is -0.6666666667y.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667y + 0.1111111112 + y2 = 0.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667y + y2 = 0.5 + 0.1111111112

Combine like terms: 0.5 + 0.1111111112 = 0.6111111112
0.1111111112 + -0.6666666667y + y2 = 0.6111111112

Factor a perfect square on the left side:
(y + -0.3333333334)(y + -0.3333333334) = 0.6111111112

Calculate the square root of the right side: 0.78173596

Break this problem into two subproblems by setting 
(y + -0.3333333334) equal to 0.78173596 and -0.78173596.


Subproblem 1
y + -0.3333333334 = 0.78173596

Simplifying
y + -0.3333333334 = 0.78173596

Reorder the terms:
-0.3333333334 + y = 0.78173596

Solving
-0.3333333334 + y = 0.78173596

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + y = 0.78173596 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + y = 0.78173596 + 0.3333333334
y = 0.78173596 + 0.3333333334

Combine like terms: 0.78173596 + 0.3333333334 = 1.1150692934
y = 1.1150692934

Simplifying
y = 1.1150692934


Subproblem 2
y + -0.3333333334 = -0.78173596

Simplifying
y + -0.3333333334 = -0.78173596

Reorder the terms:
-0.3333333334 + y = -0.78173596

Solving
-0.3333333334 + y = -0.78173596

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + y = -0.78173596 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + y = -0.78173596 + 0.3333333334
y = -0.78173596 + 0.3333333334

Combine like terms: -0.78173596 + 0.3333333334 = -0.4484026266
y = -0.4484026266

Simplifying
y = -0.4484026266


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.1150692934, -0.4484026266}

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