0=x^2-x-5

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Solution for 0=x^2-x-5 equation:


Simplifying
0 = x2 + -1x + -5

Reorder the terms:
0 = -5 + -1x + x2

Solving
0 = -5 + -1x + x2

Solving for variable 'x'.

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

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

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

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

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

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

Divide each side by '-1'.
-5 + -1x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + -1x + x2 = 0 + 5

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

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

The x term is x.  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 = 5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + -1x + x2 = 5.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

x + 0.5 = 2.291287847 Simplifying x + 0.5 = 2.291287847 Reorder the terms: 0.5 + x = 2.291287847 Solving 0.5 + x = 2.291287847 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 = 2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.291287847 + -0.5 x = 2.291287847 + -0.5 Combine like terms: 2.291287847 + -0.5 = 1.791287847 x = 1.791287847 Simplifying x = 1.791287847

Subproblem 2

x + 0.5 = -2.291287847 Simplifying x + 0.5 = -2.291287847 Reorder the terms: 0.5 + x = -2.291287847 Solving 0.5 + x = -2.291287847 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 = -2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.291287847 + -0.5 x = -2.291287847 + -0.5 Combine like terms: -2.291287847 + -0.5 = -2.791287847 x = -2.791287847 Simplifying x = -2.791287847

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.791287847, -2.791287847}

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