x^2-x-23=0

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Solution for x^2-x-23=0 equation:


Simplifying
x2 + -1x + -23 = 0

Reorder the terms:
-23 + -1x + x2 = 0

Solving
-23 + -1x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + -1x + 23 + x2 = 0 + 23

Reorder the terms:
-23 + 23 + -1x + x2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + -1x + x2 = 0 + 23
-1x + x2 = 0 + 23

Combine like terms: 0 + 23 = 23
-1x + x2 = 23

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 = 23 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 23 + 0.25

Combine like terms: 23 + 0.25 = 23.25
0.25 + -1x + x2 = 23.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 23.25

Calculate the square root of the right side: 4.82182538

Break this problem into two subproblems by setting 
(x + -0.5) equal to 4.82182538 and -4.82182538.

Subproblem 1

x + -0.5 = 4.82182538 Simplifying x + -0.5 = 4.82182538 Reorder the terms: -0.5 + x = 4.82182538 Solving -0.5 + x = 4.82182538 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 = 4.82182538 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 4.82182538 + 0.5 x = 4.82182538 + 0.5 Combine like terms: 4.82182538 + 0.5 = 5.32182538 x = 5.32182538 Simplifying x = 5.32182538

Subproblem 2

x + -0.5 = -4.82182538 Simplifying x + -0.5 = -4.82182538 Reorder the terms: -0.5 + x = -4.82182538 Solving -0.5 + x = -4.82182538 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 = -4.82182538 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -4.82182538 + 0.5 x = -4.82182538 + 0.5 Combine like terms: -4.82182538 + 0.5 = -4.32182538 x = -4.32182538 Simplifying x = -4.32182538

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.32182538, -4.32182538}

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