6y^2+3y=5

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Solution for 6y^2+3y=5 equation:


Simplifying
6y2 + 3y = 5

Reorder the terms:
3y + 6y2 = 5

Solving
3y + 6y2 = 5

Solving for variable 'y'.

Reorder the terms:
-5 + 3y + 6y2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 3y + 6y2 = 0

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

Divide each side by '6'.
-0.8333333333 + 0.5y + y2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 0.5y + 0.8333333333 + y2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 0.5y + y2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 0.5y + y2 = 0 + 0.8333333333
0.5y + y2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
0.5y + y2 = 0.8333333333

The y term is 0.5y.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5y + 0.0625 + y2 = 0.8333333333 + 0.0625

Reorder the terms:
0.0625 + 0.5y + y2 = 0.8333333333 + 0.0625

Combine like terms: 0.8333333333 + 0.0625 = 0.8958333333
0.0625 + 0.5y + y2 = 0.8958333333

Factor a perfect square on the left side:
(y + 0.25)(y + 0.25) = 0.8958333333

Calculate the square root of the right side: 0.946484724

Break this problem into two subproblems by setting 
(y + 0.25) equal to 0.946484724 and -0.946484724.

Subproblem 1

y + 0.25 = 0.946484724 Simplifying y + 0.25 = 0.946484724 Reorder the terms: 0.25 + y = 0.946484724 Solving 0.25 + y = 0.946484724 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = 0.946484724 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = 0.946484724 + -0.25 y = 0.946484724 + -0.25 Combine like terms: 0.946484724 + -0.25 = 0.696484724 y = 0.696484724 Simplifying y = 0.696484724

Subproblem 2

y + 0.25 = -0.946484724 Simplifying y + 0.25 = -0.946484724 Reorder the terms: 0.25 + y = -0.946484724 Solving 0.25 + y = -0.946484724 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = -0.946484724 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = -0.946484724 + -0.25 y = -0.946484724 + -0.25 Combine like terms: -0.946484724 + -0.25 = -1.196484724 y = -1.196484724 Simplifying y = -1.196484724

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.696484724, -1.196484724}

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