0=x^2-2x-1

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Solution for 0=x^2-2x-1 equation:


Simplifying
0 = x2 + -2x + -1

Reorder the terms:
0 = -1 + -2x + x2

Solving
0 = -1 + -2x + x2

Solving for variable 'x'.

Combine like terms: 0 + 1 = 1
1 + 2x + -1x2 = -1 + -2x + x2 + 1 + 2x + -1x2

Reorder the terms:
1 + 2x + -1x2 = -1 + 1 + -2x + 2x + x2 + -1x2

Combine like terms: -1 + 1 = 0
1 + 2x + -1x2 = 0 + -2x + 2x + x2 + -1x2
1 + 2x + -1x2 = -2x + 2x + x2 + -1x2

Combine like terms: -2x + 2x = 0
1 + 2x + -1x2 = 0 + x2 + -1x2
1 + 2x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
1 + 2x + -1x2 = 0

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

Divide each side by '-1'.
-1 + -2x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -2x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2x + x2 = 0 + 1
-2x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2x + x2 = 1

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

Add '1' to each side of the equation.
-2x + 1 + x2 = 1 + 1

Reorder the terms:
1 + -2x + x2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2x + x2 = 2

Factor a perfect square on the left side:
(x + -1)(x + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -1 = 1.414213562 Simplifying x + -1 = 1.414213562 Reorder the terms: -1 + x = 1.414213562 Solving -1 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.414213562 + 1 x = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 x = 2.414213562 Simplifying x = 2.414213562

Subproblem 2

x + -1 = -1.414213562 Simplifying x + -1 = -1.414213562 Reorder the terms: -1 + x = -1.414213562 Solving -1 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.414213562 + 1 x = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 x = -0.414213562 Simplifying x = -0.414213562

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.414213562, -0.414213562}

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