.5(y+4)(y+4)-6(y+3)=7

Solution for .5(y+4)(y+4)-6(y+3)=7 equation:

Simplifying
0.5(y + 4)(y + 4) + -6(y + 3) = 7

Reorder the terms:
0.5(4 + y)(y + 4) + -6(y + 3) = 7

Reorder the terms:
0.5(4 + y)(4 + y) + -6(y + 3) = 7

Multiply (4 + y) * (4 + y)
0.5(4(4 + y) + y(4 + y)) + -6(y + 3) = 7
0.5((4 * 4 + y * 4) + y(4 + y)) + -6(y + 3) = 7
0.5((16 + 4y) + y(4 + y)) + -6(y + 3) = 7
0.5(16 + 4y + (4 * y + y * y)) + -6(y + 3) = 7
0.5(16 + 4y + (4y + y2)) + -6(y + 3) = 7

Combine like terms: 4y + 4y = 8y
0.5(16 + 8y + y2) + -6(y + 3) = 7
(16 * 0.5 + 8y * 0.5 + y2 * 0.5) + -6(y + 3) = 7
(8 + 4y + 0.5y2) + -6(y + 3) = 7

Reorder the terms:
8 + 4y + 0.5y2 + -6(3 + y) = 7
8 + 4y + 0.5y2 + (3 * -6 + y * -6) = 7
8 + 4y + 0.5y2 + (-18 + -6y) = 7

Reorder the terms:
8 + -18 + 4y + -6y + 0.5y2 = 7

Combine like terms: 8 + -18 = -10
-10 + 4y + -6y + 0.5y2 = 7

Combine like terms: 4y + -6y = -2y
-10 + -2y + 0.5y2 = 7

Solving
-10 + -2y + 0.5y2 = 7

Solving for variable 'y'.

Reorder the terms:
-10 + -7 + -2y + 0.5y2 = 7 + -7

Combine like terms: -10 + -7 = -17
-17 + -2y + 0.5y2 = 7 + -7

Combine like terms: 7 + -7 = 0
-17 + -2y + 0.5y2 = 0

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

Divide each side by '0.5'.
-34 + -4y + y2 = 0

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + -4y + 34 + y2 = 0 + 34

Reorder the terms:
-34 + 34 + -4y + y2 = 0 + 34

Combine like terms: -34 + 34 = 0
0 + -4y + y2 = 0 + 34
-4y + y2 = 0 + 34

Combine like terms: 0 + 34 = 34
-4y + y2 = 34

The y term is -4y.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4y + 4 + y2 = 34 + 4

Reorder the terms:
4 + -4y + y2 = 34 + 4

Combine like terms: 34 + 4 = 38
4 + -4y + y2 = 38

Factor a perfect square on the left side:
(y + -2)(y + -2) = 38

Calculate the square root of the right side: 6.164414003

Break this problem into two subproblems by setting 
(y + -2) equal to 6.164414003 and -6.164414003.

Subproblem 1
y + -2 = 6.164414003

Simplifying
y + -2 = 6.164414003

Reorder the terms:
-2 + y = 6.164414003

Solving
-2 + y = 6.164414003

Solving for variable 'y'.

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

Add '2' to each side of the equation.
-2 + 2 + y = 6.164414003 + 2

Combine like terms: -2 + 2 = 0
0 + y = 6.164414003 + 2
y = 6.164414003 + 2

Combine like terms: 6.164414003 + 2 = 8.164414003
y = 8.164414003

Simplifying
y = 8.164414003

Subproblem 2
y + -2 = -6.164414003

Simplifying
y + -2 = -6.164414003

Reorder the terms:
-2 + y = -6.164414003

Solving
-2 + y = -6.164414003

Solving for variable 'y'.

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

Add '2' to each side of the equation.
-2 + 2 + y = -6.164414003 + 2

Combine like terms: -2 + 2 = 0
0 + y = -6.164414003 + 2
y = -6.164414003 + 2

Combine like terms: -6.164414003 + 2 = -4.164414003
y = -4.164414003

Simplifying
y = -4.164414003

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {8.164414003, -4.164414003}