Y^2-2Y+1=5

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Solution for Y^2-2Y+1=5 equation: 


Simplifying
Y2 + -2Y + 1 = 5

Reorder the terms:
1 + -2Y + Y2 = 5

Solving
1 + -2Y + Y2 = 5

Solving for variable 'Y'.

Reorder the terms:
1 + -5 + -2Y + Y2 = 5 + -5

Combine like terms: 1 + -5 = -4
-4 + -2Y + Y2 = 5 + -5

Combine like terms: 5 + -5 = 0
-4 + -2Y + Y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -2Y + 4 + Y2 = 0 + 4

Reorder the terms:
-4 + 4 + -2Y + Y2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -2Y + Y2 = 0 + 4
-2Y + Y2 = 0 + 4

Combine like terms: 0 + 4 = 4
-2Y + Y2 = 4

The Y term is -2Y.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2Y + 1 + Y2 = 4 + 1

Reorder the terms:
1 + -2Y + Y2 = 4 + 1

Combine like terms: 4 + 1 = 5
1 + -2Y + Y2 = 5

Factor a perfect square on the left side:
(Y + -1)(Y + -1) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(Y + -1) equal to 2.236067978 and -2.236067978.


Subproblem 1
Y + -1 = 2.236067978

Simplifying
Y + -1 = 2.236067978

Reorder the terms:
-1 + Y = 2.236067978

Solving
-1 + Y = 2.236067978

Solving for variable 'Y'.

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

Add '1' to each side of the equation.
-1 + 1 + Y = 2.236067978 + 1

Combine like terms: -1 + 1 = 0
0 + Y = 2.236067978 + 1
Y = 2.236067978 + 1

Combine like terms: 2.236067978 + 1 = 3.236067978
Y = 3.236067978

Simplifying
Y = 3.236067978


Subproblem 2
Y + -1 = -2.236067978

Simplifying
Y + -1 = -2.236067978

Reorder the terms:
-1 + Y = -2.236067978

Solving
-1 + Y = -2.236067978

Solving for variable 'Y'.

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

Add '1' to each side of the equation.
-1 + 1 + Y = -2.236067978 + 1

Combine like terms: -1 + 1 = 0
0 + Y = -2.236067978 + 1
Y = -2.236067978 + 1

Combine like terms: -2.236067978 + 1 = -1.236067978
Y = -1.236067978

Simplifying
Y = -1.236067978


Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {3.236067978, -1.236067978}

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