2y(3+y)=10

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


Simplifying
2y(3 + y) = 10
(3 * 2y + y * 2y) = 10
(6y + 2y2) = 10

Solving
6y + 2y2 = 10

Solving for variable 'y'.

Reorder the terms:
-10 + 6y + 2y2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 6y + 2y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + 3y + y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-5 + 3y + y2)' equal to zero and attempt to solve:

Simplifying
-5 + 3y + y2 = 0

Solving
-5 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 3y + 5 + y2 = 0 + 5

Reorder the terms:
-5 + 5 + 3y + y2 = 0 + 5

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

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

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 5 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + 3y + y2 = 7.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(y + 1.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
y + 1.5 = 2.692582404

Simplifying
y + 1.5 = 2.692582404

Reorder the terms:
1.5 + y = 2.692582404

Solving
1.5 + y = 2.692582404

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 2.692582404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 2.692582404 + -1.5
y = 2.692582404 + -1.5

Combine like terms: 2.692582404 + -1.5 = 1.192582404
y = 1.192582404

Simplifying
y = 1.192582404


Subproblem 2
y + 1.5 = -2.692582404

Simplifying
y + 1.5 = -2.692582404

Reorder the terms:
1.5 + y = -2.692582404

Solving
1.5 + y = -2.692582404

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -2.692582404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -2.692582404 + -1.5
y = -2.692582404 + -1.5

Combine like terms: -2.692582404 + -1.5 = -4.192582404
y = -4.192582404

Simplifying
y = -4.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.192582404, -4.192582404}
Solution
y = {1.192582404, -4.192582404}

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