y(y+12)-2(16-y)=7-y(2+y)

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Solution for y(y+12)-2(16-y)=7-y(2+y) equation:


Simplifying
y(y + 12) + -2(16 + -1y) = 7 + -1y(2 + y)

Reorder the terms:
y(12 + y) + -2(16 + -1y) = 7 + -1y(2 + y)
(12 * y + y * y) + -2(16 + -1y) = 7 + -1y(2 + y)
(12y + y2) + -2(16 + -1y) = 7 + -1y(2 + y)
12y + y2 + (16 * -2 + -1y * -2) = 7 + -1y(2 + y)
12y + y2 + (-32 + 2y) = 7 + -1y(2 + y)

Reorder the terms:
-32 + 12y + 2y + y2 = 7 + -1y(2 + y)

Combine like terms: 12y + 2y = 14y
-32 + 14y + y2 = 7 + -1y(2 + y)
-32 + 14y + y2 = 7 + (2 * -1y + y * -1y)
-32 + 14y + y2 = 7 + (-2y + -1y2)

Solving
-32 + 14y + y2 = 7 + -2y + -1y2

Solving for variable 'y'.

Reorder the terms:
-32 + -7 + 14y + 2y + y2 + y2 = 7 + -2y + -1y2 + -7 + 2y + y2

Combine like terms: -32 + -7 = -39
-39 + 14y + 2y + y2 + y2 = 7 + -2y + -1y2 + -7 + 2y + y2

Combine like terms: 14y + 2y = 16y
-39 + 16y + y2 + y2 = 7 + -2y + -1y2 + -7 + 2y + y2

Combine like terms: y2 + y2 = 2y2
-39 + 16y + 2y2 = 7 + -2y + -1y2 + -7 + 2y + y2

Reorder the terms:
-39 + 16y + 2y2 = 7 + -7 + -2y + 2y + -1y2 + y2

Combine like terms: 7 + -7 = 0
-39 + 16y + 2y2 = 0 + -2y + 2y + -1y2 + y2
-39 + 16y + 2y2 = -2y + 2y + -1y2 + y2

Combine like terms: -2y + 2y = 0
-39 + 16y + 2y2 = 0 + -1y2 + y2
-39 + 16y + 2y2 = -1y2 + y2

Combine like terms: -1y2 + y2 = 0
-39 + 16y + 2y2 = 0

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

Divide each side by '2'.
-19.5 + 8y + y2 = 0

Move the constant term to the right:

Add '19.5' to each side of the equation.
-19.5 + 8y + 19.5 + y2 = 0 + 19.5

Reorder the terms:
-19.5 + 19.5 + 8y + y2 = 0 + 19.5

Combine like terms: -19.5 + 19.5 = 0.0
0.0 + 8y + y2 = 0 + 19.5
8y + y2 = 0 + 19.5

Combine like terms: 0 + 19.5 = 19.5
8y + y2 = 19.5

The y term is 8y.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8y + 16 + y2 = 19.5 + 16

Reorder the terms:
16 + 8y + y2 = 19.5 + 16

Combine like terms: 19.5 + 16 = 35.5
16 + 8y + y2 = 35.5

Factor a perfect square on the left side:
(y + 4)(y + 4) = 35.5

Calculate the square root of the right side: 5.958187644

Break this problem into two subproblems by setting 
(y + 4) equal to 5.958187644 and -5.958187644.

Subproblem 1

y + 4 = 5.958187644 Simplifying y + 4 = 5.958187644 Reorder the terms: 4 + y = 5.958187644 Solving 4 + y = 5.958187644 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + y = 5.958187644 + -4 Combine like terms: 4 + -4 = 0 0 + y = 5.958187644 + -4 y = 5.958187644 + -4 Combine like terms: 5.958187644 + -4 = 1.958187644 y = 1.958187644 Simplifying y = 1.958187644

Subproblem 2

y + 4 = -5.958187644 Simplifying y + 4 = -5.958187644 Reorder the terms: 4 + y = -5.958187644 Solving 4 + y = -5.958187644 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + y = -5.958187644 + -4 Combine like terms: 4 + -4 = 0 0 + y = -5.958187644 + -4 y = -5.958187644 + -4 Combine like terms: -5.958187644 + -4 = -9.958187644 y = -9.958187644 Simplifying y = -9.958187644

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.958187644, -9.958187644}

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