-y^2=6y-23

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Solution for -y^2=6y-23 equation:


Simplifying
-1y2 = 6y + -23

Reorder the terms:
-1y2 = -23 + 6y

Solving
-1y2 = -23 + 6y

Solving for variable 'y'.

Reorder the terms:
23 + -6y + -1y2 = -23 + 6y + 23 + -6y

Reorder the terms:
23 + -6y + -1y2 = -23 + 23 + 6y + -6y

Combine like terms: -23 + 23 = 0
23 + -6y + -1y2 = 0 + 6y + -6y
23 + -6y + -1y2 = 6y + -6y

Combine like terms: 6y + -6y = 0
23 + -6y + -1y2 = 0

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

Divide each side by '-1'.
-23 + 6y + y2 = 0

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 6y + 23 + y2 = 0 + 23

Reorder the terms:
-23 + 23 + 6y + y2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 6y + y2 = 0 + 23
6y + y2 = 0 + 23

Combine like terms: 0 + 23 = 23
6y + y2 = 23

The y term is 6y.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6y + 9 + y2 = 23 + 9

Reorder the terms:
9 + 6y + y2 = 23 + 9

Combine like terms: 23 + 9 = 32
9 + 6y + y2 = 32

Factor a perfect square on the left side:
(y + 3)(y + 3) = 32

Calculate the square root of the right side: 5.656854249

Break this problem into two subproblems by setting 
(y + 3) equal to 5.656854249 and -5.656854249.

Subproblem 1

y + 3 = 5.656854249 Simplifying y + 3 = 5.656854249 Reorder the terms: 3 + y = 5.656854249 Solving 3 + y = 5.656854249 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + y = 5.656854249 + -3 Combine like terms: 3 + -3 = 0 0 + y = 5.656854249 + -3 y = 5.656854249 + -3 Combine like terms: 5.656854249 + -3 = 2.656854249 y = 2.656854249 Simplifying y = 2.656854249

Subproblem 2

y + 3 = -5.656854249 Simplifying y + 3 = -5.656854249 Reorder the terms: 3 + y = -5.656854249 Solving 3 + y = -5.656854249 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + y = -5.656854249 + -3 Combine like terms: 3 + -3 = 0 0 + y = -5.656854249 + -3 y = -5.656854249 + -3 Combine like terms: -5.656854249 + -3 = -8.656854249 y = -8.656854249 Simplifying y = -8.656854249

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.656854249, -8.656854249}

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