y^2-y=1

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Solution for y^2-y=1 equation: 


Simplifying
y2 + -1y = 1

Reorder the terms:
-1y + y2 = 1

Solving
-1y + y2 = 1

Solving for variable 'y'.

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 1 + 0.25

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

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1y + y2 = 1.25

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(y + -0.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
y + -0.5 = 1.118033989

Simplifying
y + -0.5 = 1.118033989

Reorder the terms:
-0.5 + y = 1.118033989

Solving
-0.5 + y = 1.118033989

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 = 1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 1.118033989 + 0.5
y = 1.118033989 + 0.5

Combine like terms: 1.118033989 + 0.5 = 1.618033989
y = 1.618033989

Simplifying
y = 1.618033989


Subproblem 2
y + -0.5 = -1.118033989

Simplifying
y + -0.5 = -1.118033989

Reorder the terms:
-0.5 + y = -1.118033989

Solving
-0.5 + y = -1.118033989

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 = -1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -1.118033989 + 0.5
y = -1.118033989 + 0.5

Combine like terms: -1.118033989 + 0.5 = -0.618033989
y = -0.618033989

Simplifying
y = -0.618033989


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.618033989, -0.618033989}

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