2y+3=y(y+7)

Solution for 2y+3=y(y+7) equation:

Simplifying
2y + 3 = y(y + 7)

Reorder the terms:
3 + 2y = y(y + 7)

Reorder the terms:
3 + 2y = y(7 + y)
3 + 2y = (7 * y + y * y)
3 + 2y = (7y + y2)

Solving
3 + 2y = 7y + y2

Solving for variable 'y'.

Combine like terms: 2y + -7y = -5y
3 + -5y + -1y2 = 7y + y2 + -7y + -1y2

Reorder the terms:
3 + -5y + -1y2 = 7y + -7y + y2 + -1y2

Combine like terms: 7y + -7y = 0
3 + -5y + -1y2 = 0 + y2 + -1y2
3 + -5y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
3 + -5y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-3 + 5y + y2 = 0

Move the constant term to the right:

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

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

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

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

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 = 3 + 6.25

Reorder the terms:
6.25 + 5y + y2 = 3 + 6.25

Combine like terms: 3 + 6.25 = 9.25
6.25 + 5y + y2 = 9.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(y + 2.5) equal to 3.041381265 and -3.041381265.

Subproblem 1
y + 2.5 = 3.041381265

Simplifying
y + 2.5 = 3.041381265

Reorder the terms:
2.5 + y = 3.041381265

Solving
2.5 + y = 3.041381265

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 = 3.041381265 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 3.041381265 + -2.5
y = 3.041381265 + -2.5

Combine like terms: 3.041381265 + -2.5 = 0.541381265
y = 0.541381265

Simplifying
y = 0.541381265

Subproblem 2
y + 2.5 = -3.041381265

Simplifying
y + 2.5 = -3.041381265

Reorder the terms:
2.5 + y = -3.041381265

Solving
2.5 + y = -3.041381265

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 = -3.041381265 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -3.041381265 + -2.5
y = -3.041381265 + -2.5

Combine like terms: -3.041381265 + -2.5 = -5.541381265
y = -5.541381265

Simplifying
y = -5.541381265

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.541381265, -5.541381265}