3y^2=3y+1

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Solution for 3y^2=3y+1 equation:


Simplifying
3y2 = 3y + 1

Reorder the terms:
3y2 = 1 + 3y

Solving
3y2 = 1 + 3y

Solving for variable 'y'.

Reorder the terms:
-1 + -3y + 3y2 = 1 + 3y + -1 + -3y

Reorder the terms:
-1 + -3y + 3y2 = 1 + -1 + 3y + -3y

Combine like terms: 1 + -1 = 0
-1 + -3y + 3y2 = 0 + 3y + -3y
-1 + -3y + 3y2 = 3y + -3y

Combine like terms: 3y + -3y = 0
-1 + -3y + 3y2 = 0

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

Divide each side by '3'.
-0.3333333333 + -1y + y2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -1y + 0.3333333333 + y2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -1y + y2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -1y + y2 = 0 + 0.3333333333
-1y + y2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-1y + y2 = 0.3333333333

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 = 0.3333333333 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
0.25 + -1y + y2 = 0.5833333333

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(y + -0.5) equal to 0.763762616 and -0.763762616.

Subproblem 1

y + -0.5 = 0.763762616 Simplifying y + -0.5 = 0.763762616 Reorder the terms: -0.5 + y = 0.763762616 Solving -0.5 + y = 0.763762616 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 = 0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 0.763762616 + 0.5 y = 0.763762616 + 0.5 Combine like terms: 0.763762616 + 0.5 = 1.263762616 y = 1.263762616 Simplifying y = 1.263762616

Subproblem 2

y + -0.5 = -0.763762616 Simplifying y + -0.5 = -0.763762616 Reorder the terms: -0.5 + y = -0.763762616 Solving -0.5 + y = -0.763762616 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 = -0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -0.763762616 + 0.5 y = -0.763762616 + 0.5 Combine like terms: -0.763762616 + 0.5 = -0.263762616 y = -0.263762616 Simplifying y = -0.263762616

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.263762616, -0.263762616}

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