3y^2+6y+1=0

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Solution for 3y^2+6y+1=0 equation:


Simplifying
3y2 + 6y + 1 = 0

Reorder the terms:
1 + 6y + 3y2 = 0

Solving
1 + 6y + 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'.
0.3333333333 + 2y + y2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 2y + -0.3333333333 + y2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 2y + y2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 2y + y2 = 0 + -0.3333333333
2y + y2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
2y + y2 = -0.3333333333

The y term is 2y.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2y + 1 + y2 = -0.3333333333 + 1

Reorder the terms:
1 + 2y + y2 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + 2y + y2 = 0.6666666667

Factor a perfect square on the left side:
(y + 1)(y + 1) = 0.6666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(y + 1) equal to 0.816496581 and -0.816496581.

Subproblem 1

y + 1 = 0.816496581 Simplifying y + 1 = 0.816496581 Reorder the terms: 1 + y = 0.816496581 Solving 1 + y = 0.816496581 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + y = 0.816496581 + -1 y = 0.816496581 + -1 Combine like terms: 0.816496581 + -1 = -0.183503419 y = -0.183503419 Simplifying y = -0.183503419

Subproblem 2

y + 1 = -0.816496581 Simplifying y + 1 = -0.816496581 Reorder the terms: 1 + y = -0.816496581 Solving 1 + y = -0.816496581 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + y = -0.816496581 + -1 y = -0.816496581 + -1 Combine like terms: -0.816496581 + -1 = -1.816496581 y = -1.816496581 Simplifying y = -1.816496581

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.183503419, -1.816496581}

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