2y(y+3)=19

Solution for 2y(y+3)=19 equation:

Simplifying
2y(y + 3) = 19

Reorder the terms:
2y(3 + y) = 19
(3 * 2y + y * 2y) = 19
(6y + 2y2) = 19

Solving
6y + 2y2 = 19

Solving for variable 'y'.

Reorder the terms:
-19 + 6y + 2y2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 6y + 2y2 = 0

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

Divide each side by '2'.
-9.5 + 3y + y2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + 3y + 9.5 + y2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + 3y + y2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + 3y + y2 = 0 + 9.5
3y + y2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
3y + y2 = 9.5

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 = 9.5 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 9.5 + 2.25

Combine like terms: 9.5 + 2.25 = 11.75
2.25 + 3y + y2 = 11.75

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 11.75

Calculate the square root of the right side: 3.4278273

Break this problem into two subproblems by setting 
(y + 1.5) equal to 3.4278273 and -3.4278273.

Subproblem 1
y + 1.5 = 3.4278273

Simplifying
y + 1.5 = 3.4278273

Reorder the terms:
1.5 + y = 3.4278273

Solving
1.5 + y = 3.4278273

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 = 3.4278273 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 3.4278273 + -1.5
y = 3.4278273 + -1.5

Combine like terms: 3.4278273 + -1.5 = 1.9278273
y = 1.9278273

Simplifying
y = 1.9278273

Subproblem 2
y + 1.5 = -3.4278273

Simplifying
y + 1.5 = -3.4278273

Reorder the terms:
1.5 + y = -3.4278273

Solving
1.5 + y = -3.4278273

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 = -3.4278273 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -3.4278273 + -1.5
y = -3.4278273 + -1.5

Combine like terms: -3.4278273 + -1.5 = -4.9278273
y = -4.9278273

Simplifying
y = -4.9278273

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.9278273, -4.9278273}