400-y^2=484-44y+y^2

Solution for 400-y^2=484-44y+y^2 equation:

Simplifying
400 + -1y2 = 484 + -44y + y2

Solving
400 + -1y2 = 484 + -44y + y2

Solving for variable 'y'.

Reorder the terms:
400 + -484 + 44y + -1y2 + -1y2 = 484 + -44y + y2 + -484 + 44y + -1y2

Combine like terms: 400 + -484 = -84
-84 + 44y + -1y2 + -1y2 = 484 + -44y + y2 + -484 + 44y + -1y2

Combine like terms: -1y2 + -1y2 = -2y2
-84 + 44y + -2y2 = 484 + -44y + y2 + -484 + 44y + -1y2

Reorder the terms:
-84 + 44y + -2y2 = 484 + -484 + -44y + 44y + y2 + -1y2

Combine like terms: 484 + -484 = 0
-84 + 44y + -2y2 = 0 + -44y + 44y + y2 + -1y2
-84 + 44y + -2y2 = -44y + 44y + y2 + -1y2

Combine like terms: -44y + 44y = 0
-84 + 44y + -2y2 = 0 + y2 + -1y2
-84 + 44y + -2y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
-84 + 44y + -2y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-42 + 22y + -1y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-42 + 22y + -1y2)' equal to zero and attempt to solve:

Simplifying
-42 + 22y + -1y2 = 0

Solving
-42 + 22y + -1y2 = 0

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

Divide each side by '-1'.
42 + -22y + y2 = 0

Move the constant term to the right:

Add '-42' to each side of the equation.
42 + -22y + -42 + y2 = 0 + -42

Reorder the terms:
42 + -42 + -22y + y2 = 0 + -42

Combine like terms: 42 + -42 = 0
0 + -22y + y2 = 0 + -42
-22y + y2 = 0 + -42

Combine like terms: 0 + -42 = -42
-22y + y2 = -42

The y term is -22y.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22y + 121 + y2 = -42 + 121

Reorder the terms:
121 + -22y + y2 = -42 + 121

Combine like terms: -42 + 121 = 79
121 + -22y + y2 = 79

Factor a perfect square on the left side:
(y + -11)(y + -11) = 79

Calculate the square root of the right side: 8.888194417

Break this problem into two subproblems by setting 
(y + -11) equal to 8.888194417 and -8.888194417.

Subproblem 1
y + -11 = 8.888194417

Simplifying
y + -11 = 8.888194417

Reorder the terms:
-11 + y = 8.888194417

Solving
-11 + y = 8.888194417

Solving for variable 'y'.

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

Add '11' to each side of the equation.
-11 + 11 + y = 8.888194417 + 11

Combine like terms: -11 + 11 = 0
0 + y = 8.888194417 + 11
y = 8.888194417 + 11

Combine like terms: 8.888194417 + 11 = 19.888194417
y = 19.888194417

Simplifying
y = 19.888194417

Subproblem 2
y + -11 = -8.888194417

Simplifying
y + -11 = -8.888194417

Reorder the terms:
-11 + y = -8.888194417

Solving
-11 + y = -8.888194417

Solving for variable 'y'.

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

Add '11' to each side of the equation.
-11 + 11 + y = -8.888194417 + 11

Combine like terms: -11 + 11 = 0
0 + y = -8.888194417 + 11
y = -8.888194417 + 11

Combine like terms: -8.888194417 + 11 = 2.111805583
y = 2.111805583

Simplifying
y = 2.111805583

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {19.888194417, 2.111805583}
Solution
y = {19.888194417, 2.111805583}