x^2+x=499999999999500000

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Solution for x^2+x=499999999999500000 equation:


Simplifying
x2 + x = 499999999999500000

Reorder the terms:
x + x2 = 499999999999500000

Solving
x + x2 = 499999999999500000

Solving for variable 'x'.

Reorder the terms:
-499999999999500000 + x + x2 = 499999999999500000 + -499999999999500000

Combine like terms: 499999999999500000 + -499999999999500000 = 0
-499999999999500000 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '499999999999500000' to each side of the equation.
-499999999999500000 + x + 499999999999500000 + x2 = 0 + 499999999999500000

Reorder the terms:
-499999999999500000 + 499999999999500000 + x + x2 = 0 + 499999999999500000

Combine like terms: -499999999999500000 + 499999999999500000 = 0
0 + x + x2 = 0 + 499999999999500000
x + x2 = 0 + 499999999999500000

Combine like terms: 0 + 499999999999500000 = 499999999999500000
x + x2 = 499999999999500000

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.
x + 0.25 + x2 = 499999999999500000 + 0.25

Reorder the terms:
0.25 + x + x2 = 499999999999500000 + 0.25

Combine like terms: 499999999999500000 + 0.25 = 499999999999500000.25
0.25 + x + x2 = 499999999999500000.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 499999999999500000.25

Calculate the square root of the right side: 707106781.186193971

Break this problem into two subproblems by setting 
(x + 0.5) equal to 707106781.186193971 and -707106781.186193971.

Subproblem 1

x + 0.5 = 707106781.186193971 Simplifying x + 0.5 = 707106781.186193971 Reorder the terms: 0.5 + x = 707106781.186193971 Solving 0.5 + x = 707106781.186193971 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 = 707106781.186193971 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 707106781.186193971 + -0.5 x = 707106781.186193971 + -0.5 Combine like terms: 707106781.186193971 + -0.5 = 707106780.686193971 x = 707106780.686193971 Simplifying x = 707106780.686193971

Subproblem 2

x + 0.5 = -707106781.186193971 Simplifying x + 0.5 = -707106781.186193971 Reorder the terms: 0.5 + x = -707106781.186193971 Solving 0.5 + x = -707106781.186193971 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 = -707106781.186193971 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -707106781.186193971 + -0.5 x = -707106781.186193971 + -0.5 Combine like terms: -707106781.186193971 + -0.5 = -707106781.686193971 x = -707106781.686193971 Simplifying x = -707106781.686193971

Solution

The solution to the problem is based on the solutions from the subproblems. x = {707106780.686193971, -707106781.686193971}

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