3y-6y^2+4=0

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


Simplifying
3y + -6y2 + 4 = 0

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

Solving
4 + 3y + -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'.
-0.6666666667 + -0.5y + y2 = 0

Move the constant term to the right:

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

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

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -0.5y + y2 = 0 + 0.6666666667
-0.5y + y2 = 0 + 0.6666666667

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

The y term is -0.5y.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5y + 0.0625 + y2 = 0.6666666667 + 0.0625

Reorder the terms:
0.0625 + -0.5y + y2 = 0.6666666667 + 0.0625

Combine like terms: 0.6666666667 + 0.0625 = 0.7291666667
0.0625 + -0.5y + y2 = 0.7291666667

Factor a perfect square on the left side:
(y + -0.25)(y + -0.25) = 0.7291666667

Calculate the square root of the right side: 0.853912564

Break this problem into two subproblems by setting 
(y + -0.25) equal to 0.853912564 and -0.853912564.

Subproblem 1

y + -0.25 = 0.853912564 Simplifying y + -0.25 = 0.853912564 Reorder the terms: -0.25 + y = 0.853912564 Solving -0.25 + y = 0.853912564 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = 0.853912564 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = 0.853912564 + 0.25 y = 0.853912564 + 0.25 Combine like terms: 0.853912564 + 0.25 = 1.103912564 y = 1.103912564 Simplifying y = 1.103912564

Subproblem 2

y + -0.25 = -0.853912564 Simplifying y + -0.25 = -0.853912564 Reorder the terms: -0.25 + y = -0.853912564 Solving -0.25 + y = -0.853912564 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = -0.853912564 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = -0.853912564 + 0.25 y = -0.853912564 + 0.25 Combine like terms: -0.853912564 + 0.25 = -0.603912564 y = -0.603912564 Simplifying y = -0.603912564

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.103912564, -0.603912564}

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