3y(y+5)=45

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Solution for 3y(y+5)=45 equation: 


Simplifying
3y(y + 5) = 45

Reorder the terms:
3y(5 + y) = 45
(5 * 3y + y * 3y) = 45
(15y + 3y2) = 45

Solving
15y + 3y2 = 45

Solving for variable 'y'.

Reorder the terms:
-45 + 15y + 3y2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 15y + 3y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-15 + 5y + y2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-15 + 5y + y2)' equal to zero and attempt to solve:

Simplifying
-15 + 5y + y2 = 0

Solving
-15 + 5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 5y + 15 + y2 = 0 + 15

Reorder the terms:
-15 + 15 + 5y + y2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 5y + y2 = 0 + 15
5y + y2 = 0 + 15

Combine like terms: 0 + 15 = 15
5y + y2 = 15

The y term is 5y.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5y + 6.25 + y2 = 15 + 6.25

Reorder the terms:
6.25 + 5y + y2 = 15 + 6.25

Combine like terms: 15 + 6.25 = 21.25
6.25 + 5y + y2 = 21.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(y + 2.5) equal to 4.609772229 and -4.609772229.


Subproblem 1
y + 2.5 = 4.609772229

Simplifying
y + 2.5 = 4.609772229

Reorder the terms:
2.5 + y = 4.609772229

Solving
2.5 + y = 4.609772229

Solving for variable 'y'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = 4.609772229 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 4.609772229 + -2.5
y = 4.609772229 + -2.5

Combine like terms: 4.609772229 + -2.5 = 2.109772229
y = 2.109772229

Simplifying
y = 2.109772229


Subproblem 2
y + 2.5 = -4.609772229

Simplifying
y + 2.5 = -4.609772229

Reorder the terms:
2.5 + y = -4.609772229

Solving
2.5 + y = -4.609772229

Solving for variable 'y'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = -4.609772229 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -4.609772229 + -2.5
y = -4.609772229 + -2.5

Combine like terms: -4.609772229 + -2.5 = -7.109772229
y = -7.109772229

Simplifying
y = -7.109772229


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.109772229, -7.109772229}
Solution
y = {2.109772229, -7.109772229}

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