3y^2+9y+2=0

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Solution for 3y^2+9y+2=0 equation:


Simplifying
3y2 + 9y + 2 = 0

Reorder the terms:
2 + 9y + 3y2 = 0

Solving
2 + 9y + 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 + 3y + y2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 3y + -0.6666666667 + y2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 3y + y2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 3y + y2 = 0 + -0.6666666667
3y + y2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
3y + y2 = -0.6666666667

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = -0.6666666667 + 2.25

Reorder the terms:
2.25 + 3y + y2 = -0.6666666667 + 2.25

Combine like terms: -0.6666666667 + 2.25 = 1.5833333333
2.25 + 3y + y2 = 1.5833333333

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 1.5833333333

Calculate the square root of the right side: 1.258305739

Break this problem into two subproblems by setting 
(y + 1.5) equal to 1.258305739 and -1.258305739.

Subproblem 1

y + 1.5 = 1.258305739 Simplifying y + 1.5 = 1.258305739 Reorder the terms: 1.5 + y = 1.258305739 Solving 1.5 + y = 1.258305739 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 1.258305739 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 1.258305739 + -1.5 y = 1.258305739 + -1.5 Combine like terms: 1.258305739 + -1.5 = -0.241694261 y = -0.241694261 Simplifying y = -0.241694261

Subproblem 2

y + 1.5 = -1.258305739 Simplifying y + 1.5 = -1.258305739 Reorder the terms: 1.5 + y = -1.258305739 Solving 1.5 + y = -1.258305739 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -1.258305739 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -1.258305739 + -1.5 y = -1.258305739 + -1.5 Combine like terms: -1.258305739 + -1.5 = -2.758305739 y = -2.758305739 Simplifying y = -2.758305739

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.241694261, -2.758305739}

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