5x^2-x-5=0

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


Simplifying
5x2 + -1x + -5 = 0

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

Solving
-5 + -1x + 5x2 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
-1 + -0.2x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.2x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.2x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.2x + x2 = 0 + 1
-0.2x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.2x + x2 = 1

The x term is -0.2x.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2x + 0.01 + x2 = 1 + 0.01

Reorder the terms:
0.01 + -0.2x + x2 = 1 + 0.01

Combine like terms: 1 + 0.01 = 1.01
0.01 + -0.2x + x2 = 1.01

Factor a perfect square on the left side:
(x + -0.1)(x + -0.1) = 1.01

Calculate the square root of the right side: 1.004987562

Break this problem into two subproblems by setting 
(x + -0.1) equal to 1.004987562 and -1.004987562.

Subproblem 1

x + -0.1 = 1.004987562 Simplifying x + -0.1 = 1.004987562 Reorder the terms: -0.1 + x = 1.004987562 Solving -0.1 + x = 1.004987562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 1.004987562 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 1.004987562 + 0.1 x = 1.004987562 + 0.1 Combine like terms: 1.004987562 + 0.1 = 1.104987562 x = 1.104987562 Simplifying x = 1.104987562

Subproblem 2

x + -0.1 = -1.004987562 Simplifying x + -0.1 = -1.004987562 Reorder the terms: -0.1 + x = -1.004987562 Solving -0.1 + x = -1.004987562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -1.004987562 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -1.004987562 + 0.1 x = -1.004987562 + 0.1 Combine like terms: -1.004987562 + 0.1 = -0.904987562 x = -0.904987562 Simplifying x = -0.904987562

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.104987562, -0.904987562}

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