3y^2+10y+6=0

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Solution for 3y^2+10y+6=0 equation:


Simplifying
3y2 + 10y + 6 = 0

Reorder the terms:
6 + 10y + 3y2 = 0

Solving
6 + 10y + 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'.
2 + 3.333333333y + y2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 3.333333333y + -2 + y2 = 0 + -2

Reorder the terms:
2 + -2 + 3.333333333y + y2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 3.333333333y + y2 = 0 + -2
3.333333333y + y2 = 0 + -2

Combine like terms: 0 + -2 = -2
3.333333333y + y2 = -2

The y term is 3.333333333y.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333y + 2.777777779 + y2 = -2 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333y + y2 = -2 + 2.777777779

Combine like terms: -2 + 2.777777779 = 0.777777779
2.777777779 + 3.333333333y + y2 = 0.777777779

Factor a perfect square on the left side:
(y + 1.666666667)(y + 1.666666667) = 0.777777779

Calculate the square root of the right side: 0.881917104

Break this problem into two subproblems by setting 
(y + 1.666666667) equal to 0.881917104 and -0.881917104.

Subproblem 1

y + 1.666666667 = 0.881917104 Simplifying y + 1.666666667 = 0.881917104 Reorder the terms: 1.666666667 + y = 0.881917104 Solving 1.666666667 + y = 0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = 0.881917104 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = 0.881917104 + -1.666666667 y = 0.881917104 + -1.666666667 Combine like terms: 0.881917104 + -1.666666667 = -0.784749563 y = -0.784749563 Simplifying y = -0.784749563

Subproblem 2

y + 1.666666667 = -0.881917104 Simplifying y + 1.666666667 = -0.881917104 Reorder the terms: 1.666666667 + y = -0.881917104 Solving 1.666666667 + y = -0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = -0.881917104 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = -0.881917104 + -1.666666667 y = -0.881917104 + -1.666666667 Combine like terms: -0.881917104 + -1.666666667 = -2.548583771 y = -2.548583771 Simplifying y = -2.548583771

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.784749563, -2.548583771}

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