(3y+1)*8y=23

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Solution for (3y+1)*8y=23 equation:


Simplifying
(3y + 1) * 8y = 23

Reorder the terms:
(1 + 3y) * 8y = 23

Reorder the terms for easier multiplication:
8y(1 + 3y) = 23
(1 * 8y + 3y * 8y) = 23
(8y + 24y2) = 23

Solving
8y + 24y2 = 23

Solving for variable 'y'.

Reorder the terms:
-23 + 8y + 24y2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + 8y + 24y2 = 0

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

Divide each side by '24'.
-0.9583333333 + 0.3333333333y + y2 = 0

Move the constant term to the right:

Add '0.9583333333' to each side of the equation.
-0.9583333333 + 0.3333333333y + 0.9583333333 + y2 = 0 + 0.9583333333

Reorder the terms:
-0.9583333333 + 0.9583333333 + 0.3333333333y + y2 = 0 + 0.9583333333

Combine like terms: -0.9583333333 + 0.9583333333 = 0.0000000000
0.0000000000 + 0.3333333333y + y2 = 0 + 0.9583333333
0.3333333333y + y2 = 0 + 0.9583333333

Combine like terms: 0 + 0.9583333333 = 0.9583333333
0.3333333333y + y2 = 0.9583333333

The y term is 0.3333333333y.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333y + 0.02777777779 + y2 = 0.9583333333 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333y + y2 = 0.9583333333 + 0.02777777779

Combine like terms: 0.9583333333 + 0.02777777779 = 0.98611111109
0.02777777779 + 0.3333333333y + y2 = 0.98611111109

Factor a perfect square on the left side:
(y + 0.1666666667)(y + 0.1666666667) = 0.98611111109

Calculate the square root of the right side: 0.993031274

Break this problem into two subproblems by setting 
(y + 0.1666666667) equal to 0.993031274 and -0.993031274.

Subproblem 1

y + 0.1666666667 = 0.993031274 Simplifying y + 0.1666666667 = 0.993031274 Reorder the terms: 0.1666666667 + y = 0.993031274 Solving 0.1666666667 + y = 0.993031274 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = 0.993031274 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = 0.993031274 + -0.1666666667 y = 0.993031274 + -0.1666666667 Combine like terms: 0.993031274 + -0.1666666667 = 0.8263646073 y = 0.8263646073 Simplifying y = 0.8263646073

Subproblem 2

y + 0.1666666667 = -0.993031274 Simplifying y + 0.1666666667 = -0.993031274 Reorder the terms: 0.1666666667 + y = -0.993031274 Solving 0.1666666667 + y = -0.993031274 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = -0.993031274 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = -0.993031274 + -0.1666666667 y = -0.993031274 + -0.1666666667 Combine like terms: -0.993031274 + -0.1666666667 = -1.1596979407 y = -1.1596979407 Simplifying y = -1.1596979407

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.8263646073, -1.1596979407}

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