27=9(y-3)+3y+y^2

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Solution for 27=9(y-3)+3y+y^2 equation:


Simplifying
27 = 9(y + -3) + 3y + y2

Reorder the terms:
27 = 9(-3 + y) + 3y + y2
27 = (-3 * 9 + y * 9) + 3y + y2
27 = (-27 + 9y) + 3y + y2

Combine like terms: 9y + 3y = 12y
27 = -27 + 12y + y2

Solving
27 = -27 + 12y + y2

Solving for variable 'y'.

Combine like terms: 27 + 27 = 54
54 + -12y + -1y2 = -27 + 12y + y2 + 27 + -12y + -1y2

Reorder the terms:
54 + -12y + -1y2 = -27 + 27 + 12y + -12y + y2 + -1y2

Combine like terms: -27 + 27 = 0
54 + -12y + -1y2 = 0 + 12y + -12y + y2 + -1y2
54 + -12y + -1y2 = 12y + -12y + y2 + -1y2

Combine like terms: 12y + -12y = 0
54 + -12y + -1y2 = 0 + y2 + -1y2
54 + -12y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
54 + -12y + -1y2 = 0

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

Divide each side by '-1'.
-54 + 12y + y2 = 0

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + 12y + 54 + y2 = 0 + 54

Reorder the terms:
-54 + 54 + 12y + y2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + 12y + y2 = 0 + 54
12y + y2 = 0 + 54

Combine like terms: 0 + 54 = 54
12y + y2 = 54

The y term is 12y.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12y + 36 + y2 = 54 + 36

Reorder the terms:
36 + 12y + y2 = 54 + 36

Combine like terms: 54 + 36 = 90
36 + 12y + y2 = 90

Factor a perfect square on the left side:
(y + 6)(y + 6) = 90

Calculate the square root of the right side: 9.486832981

Break this problem into two subproblems by setting 
(y + 6) equal to 9.486832981 and -9.486832981.

Subproblem 1

y + 6 = 9.486832981 Simplifying y + 6 = 9.486832981 Reorder the terms: 6 + y = 9.486832981 Solving 6 + y = 9.486832981 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + y = 9.486832981 + -6 Combine like terms: 6 + -6 = 0 0 + y = 9.486832981 + -6 y = 9.486832981 + -6 Combine like terms: 9.486832981 + -6 = 3.486832981 y = 3.486832981 Simplifying y = 3.486832981

Subproblem 2

y + 6 = -9.486832981 Simplifying y + 6 = -9.486832981 Reorder the terms: 6 + y = -9.486832981 Solving 6 + y = -9.486832981 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + y = -9.486832981 + -6 Combine like terms: 6 + -6 = 0 0 + y = -9.486832981 + -6 y = -9.486832981 + -6 Combine like terms: -9.486832981 + -6 = -15.486832981 y = -15.486832981 Simplifying y = -15.486832981

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.486832981, -15.486832981}

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