(x^2-x-1)=5+x

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Solution for (x^2-x-1)=5+x equation:


Simplifying
(x2 + -1x + -1) = 5 + x

Reorder the terms:
(-1 + -1x + x2) = 5 + x

Remove parenthesis around (-1 + -1x + x2)
-1 + -1x + x2 = 5 + x

Solving
-1 + -1x + x2 = 5 + x

Solving for variable 'x'.

Reorder the terms:
-1 + -5 + -1x + -1x + x2 = 5 + x + -5 + -1x

Combine like terms: -1 + -5 = -6
-6 + -1x + -1x + x2 = 5 + x + -5 + -1x

Combine like terms: -1x + -1x = -2x
-6 + -2x + x2 = 5 + x + -5 + -1x

Reorder the terms:
-6 + -2x + x2 = 5 + -5 + x + -1x

Combine like terms: 5 + -5 = 0
-6 + -2x + x2 = 0 + x + -1x
-6 + -2x + x2 = x + -1x

Combine like terms: x + -1x = 0
-6 + -2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -2x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + -2x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -2x + x2 = 0 + 6
-2x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
-2x + x2 = 6

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 = 6 + 1

Reorder the terms:
1 + -2x + x2 = 6 + 1

Combine like terms: 6 + 1 = 7
1 + -2x + x2 = 7

Factor a perfect square on the left side:
(x + -1)(x + -1) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(x + -1) equal to 2.645751311 and -2.645751311.

Subproblem 1

x + -1 = 2.645751311 Simplifying x + -1 = 2.645751311 Reorder the terms: -1 + x = 2.645751311 Solving -1 + x = 2.645751311 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 = 2.645751311 + 1 Combine like terms: -1 + 1 = 0 0 + x = 2.645751311 + 1 x = 2.645751311 + 1 Combine like terms: 2.645751311 + 1 = 3.645751311 x = 3.645751311 Simplifying x = 3.645751311

Subproblem 2

x + -1 = -2.645751311 Simplifying x + -1 = -2.645751311 Reorder the terms: -1 + x = -2.645751311 Solving -1 + x = -2.645751311 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 = -2.645751311 + 1 Combine like terms: -1 + 1 = 0 0 + x = -2.645751311 + 1 x = -2.645751311 + 1 Combine like terms: -2.645751311 + 1 = -1.645751311 x = -1.645751311 Simplifying x = -1.645751311

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.645751311, -1.645751311}

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