x^2-x-1=3

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Solution for x^2-x-1=3 equation:


Simplifying
x2 + -1x + -1 = 3

Reorder the terms:
-1 + -1x + x2 = 3

Solving
-1 + -1x + x2 = 3

Solving for variable 'x'.

Reorder the terms:
-1 + -3 + -1x + x2 = 3 + -3

Combine like terms: -1 + -3 = -4
-4 + -1x + x2 = 3 + -3

Combine like terms: 3 + -3 = 0
-4 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -1x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -1x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -1x + x2 = 0 + 4
-1x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-1x + x2 = 4

The x term is -1x.  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.
-1x + 0.25 + x2 = 4 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + -1x + x2 = 4.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

x + -0.5 = 2.061552813 Simplifying x + -0.5 = 2.061552813 Reorder the terms: -0.5 + x = 2.061552813 Solving -0.5 + x = 2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 2.061552813 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 2.061552813 + 0.5 x = 2.061552813 + 0.5 Combine like terms: 2.061552813 + 0.5 = 2.561552813 x = 2.561552813 Simplifying x = 2.561552813

Subproblem 2

x + -0.5 = -2.061552813 Simplifying x + -0.5 = -2.061552813 Reorder the terms: -0.5 + x = -2.061552813 Solving -0.5 + x = -2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -2.061552813 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -2.061552813 + 0.5 x = -2.061552813 + 0.5 Combine like terms: -2.061552813 + 0.5 = -1.561552813 x = -1.561552813 Simplifying x = -1.561552813

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.561552813, -1.561552813}

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