0=x^2-5x-1

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Solution for 0=x^2-5x-1 equation:


Simplifying
0 = x2 + -5x + -1

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

Solving
0 = -1 + -5x + x2

Solving for variable 'x'.

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

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

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

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

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

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

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

Move the constant term to the right:

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

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

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

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

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 1 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 1 + 6.25

Combine like terms: 1 + 6.25 = 7.25
6.25 + -5x + x2 = 7.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

x + -2.5 = 2.692582404 Simplifying x + -2.5 = 2.692582404 Reorder the terms: -2.5 + x = 2.692582404 Solving -2.5 + x = 2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 2.692582404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.692582404 + 2.5 x = 2.692582404 + 2.5 Combine like terms: 2.692582404 + 2.5 = 5.192582404 x = 5.192582404 Simplifying x = 5.192582404

Subproblem 2

x + -2.5 = -2.692582404 Simplifying x + -2.5 = -2.692582404 Reorder the terms: -2.5 + x = -2.692582404 Solving -2.5 + x = -2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -2.692582404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.692582404 + 2.5 x = -2.692582404 + 2.5 Combine like terms: -2.692582404 + 2.5 = -0.192582404 x = -0.192582404 Simplifying x = -0.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.192582404, -0.192582404}

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