y^2-4y+2=1

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Solution for y^2-4y+2=1 equation: 


Simplifying
y2 + -4y + 2 = 1

Reorder the terms:
2 + -4y + y2 = 1

Solving
2 + -4y + y2 = 1

Solving for variable 'y'.

Reorder the terms:
2 + -1 + -4y + y2 = 1 + -1

Combine like terms: 2 + -1 = 1
1 + -4y + y2 = 1 + -1

Combine like terms: 1 + -1 = 0
1 + -4y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4y + -1 + y2 = 0 + -1

Reorder the terms:
1 + -1 + -4y + y2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4y + y2 = 0 + -1
-4y + y2 = 0 + -1

Combine like terms: 0 + -1 = -1
-4y + y2 = -1

The y term is -4y.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4y + 4 + y2 = -1 + 4

Reorder the terms:
4 + -4y + y2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + -4y + y2 = 3

Factor a perfect square on the left side:
(y + -2)(y + -2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(y + -2) equal to 1.732050808 and -1.732050808.


Subproblem 1
y + -2 = 1.732050808

Simplifying
y + -2 = 1.732050808

Reorder the terms:
-2 + y = 1.732050808

Solving
-2 + y = 1.732050808

Solving for variable 'y'.

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

Add '2' to each side of the equation.
-2 + 2 + y = 1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + y = 1.732050808 + 2
y = 1.732050808 + 2

Combine like terms: 1.732050808 + 2 = 3.732050808
y = 3.732050808

Simplifying
y = 3.732050808


Subproblem 2
y + -2 = -1.732050808

Simplifying
y + -2 = -1.732050808

Reorder the terms:
-2 + y = -1.732050808

Solving
-2 + y = -1.732050808

Solving for variable 'y'.

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

Add '2' to each side of the equation.
-2 + 2 + y = -1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + y = -1.732050808 + 2
y = -1.732050808 + 2

Combine like terms: -1.732050808 + 2 = 0.267949192
y = 0.267949192

Simplifying
y = 0.267949192


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.732050808, 0.267949192}

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