3y^2+7y+3=0

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Solution for 3y^2+7y+3=0 equation:


Simplifying
3y2 + 7y + 3 = 0

Reorder the terms:
3 + 7y + 3y2 = 0

Solving
3 + 7y + 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'.
1 + 2.333333333y + y2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.333333333y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + 2.333333333y + y2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.333333333y + y2 = 0 + -1
2.333333333y + y2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.333333333y + y2 = -1

The y term is 2.333333333y.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333y + 1.361111112 + y2 = -1 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333y + y2 = -1 + 1.361111112

Combine like terms: -1 + 1.361111112 = 0.361111112
1.361111112 + 2.333333333y + y2 = 0.361111112

Factor a perfect square on the left side:
(y + 1.166666667)(y + 1.166666667) = 0.361111112

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(y + 1.166666667) equal to 0.600925213 and -0.600925213.

Subproblem 1

y + 1.166666667 = 0.600925213 Simplifying y + 1.166666667 = 0.600925213 Reorder the terms: 1.166666667 + y = 0.600925213 Solving 1.166666667 + y = 0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + y = 0.600925213 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + y = 0.600925213 + -1.166666667 y = 0.600925213 + -1.166666667 Combine like terms: 0.600925213 + -1.166666667 = -0.565741454 y = -0.565741454 Simplifying y = -0.565741454

Subproblem 2

y + 1.166666667 = -0.600925213 Simplifying y + 1.166666667 = -0.600925213 Reorder the terms: 1.166666667 + y = -0.600925213 Solving 1.166666667 + y = -0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + y = -0.600925213 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + y = -0.600925213 + -1.166666667 y = -0.600925213 + -1.166666667 Combine like terms: -0.600925213 + -1.166666667 = -1.76759188 y = -1.76759188 Simplifying y = -1.76759188

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.565741454, -1.76759188}

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