3y(2y+3)=10

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Solution for 3y(2y+3)=10 equation:


Simplifying
3y(2y + 3) = 10

Reorder the terms:
3y(3 + 2y) = 10
(3 * 3y + 2y * 3y) = 10
(9y + 6y2) = 10

Solving
9y + 6y2 = 10

Solving for variable 'y'.

Reorder the terms:
-10 + 9y + 6y2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 9y + 6y2 = 0

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

Divide each side by '6'.
-1.666666667 + 1.5y + y2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.5y + 1.666666667 + y2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 1.5y + y2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 1.5y + y2 = 0 + 1.666666667
1.5y + y2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
1.5y + y2 = 1.666666667

The y term is 1.5y.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5y + 0.5625 + y2 = 1.666666667 + 0.5625

Reorder the terms:
0.5625 + 1.5y + y2 = 1.666666667 + 0.5625

Combine like terms: 1.666666667 + 0.5625 = 2.229166667
0.5625 + 1.5y + y2 = 2.229166667

Factor a perfect square on the left side:
(y + 0.75)(y + 0.75) = 2.229166667

Calculate the square root of the right side: 1.493039406

Break this problem into two subproblems by setting 
(y + 0.75) equal to 1.493039406 and -1.493039406.

Subproblem 1

y + 0.75 = 1.493039406 Simplifying y + 0.75 = 1.493039406 Reorder the terms: 0.75 + y = 1.493039406 Solving 0.75 + y = 1.493039406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 1.493039406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 1.493039406 + -0.75 y = 1.493039406 + -0.75 Combine like terms: 1.493039406 + -0.75 = 0.743039406 y = 0.743039406 Simplifying y = 0.743039406

Subproblem 2

y + 0.75 = -1.493039406 Simplifying y + 0.75 = -1.493039406 Reorder the terms: 0.75 + y = -1.493039406 Solving 0.75 + y = -1.493039406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -1.493039406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -1.493039406 + -0.75 y = -1.493039406 + -0.75 Combine like terms: -1.493039406 + -0.75 = -2.243039406 y = -2.243039406 Simplifying y = -2.243039406

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.743039406, -2.243039406}

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