y^2-54=y

Solution for y^2-54=y equation:

Simplifying
y2 + -54 = y

Reorder the terms:
-54 + y2 = y

Solving
-54 + y2 = y

Solving for variable 'y'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -1y + 54 + y2 = 0 + 54

Reorder the terms:
-54 + 54 + -1y + y2 = 0 + 54

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

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

The y term is -1y.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1y + 0.25 + y2 = 54 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 54 + 0.25

Combine like terms: 54 + 0.25 = 54.25
0.25 + -1y + y2 = 54.25

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 54.25

Calculate the square root of the right side: 7.365459931

Break this problem into two subproblems by setting 
(y + -0.5) equal to 7.365459931 and -7.365459931.

Subproblem 1
y + -0.5 = 7.365459931

Simplifying
y + -0.5 = 7.365459931

Reorder the terms:
-0.5 + y = 7.365459931

Solving
-0.5 + y = 7.365459931

Solving for variable 'y'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + y = 7.365459931 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 7.365459931 + 0.5
y = 7.365459931 + 0.5

Combine like terms: 7.365459931 + 0.5 = 7.865459931
y = 7.865459931

Simplifying
y = 7.865459931

Subproblem 2
y + -0.5 = -7.365459931

Simplifying
y + -0.5 = -7.365459931

Reorder the terms:
-0.5 + y = -7.365459931

Solving
-0.5 + y = -7.365459931

Solving for variable 'y'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + y = -7.365459931 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -7.365459931 + 0.5
y = -7.365459931 + 0.5

Combine like terms: -7.365459931 + 0.5 = -6.865459931
y = -6.865459931

Simplifying
y = -6.865459931

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {7.865459931, -6.865459931}