3y^2+5y=3

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Solution for 3y^2+5y=3 equation:


Simplifying
3y2 + 5y = 3

Reorder the terms:
5y + 3y2 = 3

Solving
5y + 3y2 = 3

Solving for variable 'y'.

Reorder the terms:
-3 + 5y + 3y2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + 5y + 3y2 = 0

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

Divide each side by '3'.
-1 + 1.666666667y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.666666667y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.666666667y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.666666667y + y2 = 0 + 1
1.666666667y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.666666667y + y2 = 1

The y term is 1.666666667y.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667y + 0.6944444447 + y2 = 1 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667y + y2 = 1 + 0.6944444447

Combine like terms: 1 + 0.6944444447 = 1.6944444447
0.6944444447 + 1.666666667y + y2 = 1.6944444447

Factor a perfect square on the left side:
(y + 0.8333333335)(y + 0.8333333335) = 1.6944444447

Calculate the square root of the right side: 1.301708279

Break this problem into two subproblems by setting 
(y + 0.8333333335) equal to 1.301708279 and -1.301708279.

Subproblem 1

y + 0.8333333335 = 1.301708279 Simplifying y + 0.8333333335 = 1.301708279 Reorder the terms: 0.8333333335 + y = 1.301708279 Solving 0.8333333335 + y = 1.301708279 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = 1.301708279 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = 1.301708279 + -0.8333333335 y = 1.301708279 + -0.8333333335 Combine like terms: 1.301708279 + -0.8333333335 = 0.4683749455 y = 0.4683749455 Simplifying y = 0.4683749455

Subproblem 2

y + 0.8333333335 = -1.301708279 Simplifying y + 0.8333333335 = -1.301708279 Reorder the terms: 0.8333333335 + y = -1.301708279 Solving 0.8333333335 + y = -1.301708279 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = -1.301708279 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = -1.301708279 + -0.8333333335 y = -1.301708279 + -0.8333333335 Combine like terms: -1.301708279 + -0.8333333335 = -2.1350416125 y = -2.1350416125 Simplifying y = -2.1350416125

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.4683749455, -2.1350416125}

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