2x^2-x-500=0

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Solution for 2x^2-x-500=0 equation:


Simplifying
2x2 + -1x + -500 = 0

Reorder the terms:
-500 + -1x + 2x2 = 0

Solving
-500 + -1x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-250 + -0.5x + x2 = 0

Move the constant term to the right:

Add '250' to each side of the equation.
-250 + -0.5x + 250 + x2 = 0 + 250

Reorder the terms:
-250 + 250 + -0.5x + x2 = 0 + 250

Combine like terms: -250 + 250 = 0
0 + -0.5x + x2 = 0 + 250
-0.5x + x2 = 0 + 250

Combine like terms: 0 + 250 = 250
-0.5x + x2 = 250

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 250 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 250 + 0.0625

Combine like terms: 250 + 0.0625 = 250.0625
0.0625 + -0.5x + x2 = 250.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 250.0625

Calculate the square root of the right side: 15.813364601

Break this problem into two subproblems by setting 
(x + -0.25) equal to 15.813364601 and -15.813364601.

Subproblem 1

x + -0.25 = 15.813364601 Simplifying x + -0.25 = 15.813364601 Reorder the terms: -0.25 + x = 15.813364601 Solving -0.25 + x = 15.813364601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 15.813364601 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 15.813364601 + 0.25 x = 15.813364601 + 0.25 Combine like terms: 15.813364601 + 0.25 = 16.063364601 x = 16.063364601 Simplifying x = 16.063364601

Subproblem 2

x + -0.25 = -15.813364601 Simplifying x + -0.25 = -15.813364601 Reorder the terms: -0.25 + x = -15.813364601 Solving -0.25 + x = -15.813364601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -15.813364601 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -15.813364601 + 0.25 x = -15.813364601 + 0.25 Combine like terms: -15.813364601 + 0.25 = -15.563364601 x = -15.563364601 Simplifying x = -15.563364601

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16.063364601, -15.563364601}

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