x^2+2x=1000

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Solution for x^2+2x=1000 equation:


Simplifying
x2 + 2x = 1000

Reorder the terms:
2x + x2 = 1000

Solving
2x + x2 = 1000

Solving for variable 'x'.

Reorder the terms:
-1000 + 2x + x2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1000' to each side of the equation.
-1000 + 2x + 1000 + x2 = 0 + 1000

Reorder the terms:
-1000 + 1000 + 2x + x2 = 0 + 1000

Combine like terms: -1000 + 1000 = 0
0 + 2x + x2 = 0 + 1000
2x + x2 = 0 + 1000

Combine like terms: 0 + 1000 = 1000
2x + x2 = 1000

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 1000 + 1

Reorder the terms:
1 + 2x + x2 = 1000 + 1

Combine like terms: 1000 + 1 = 1001
1 + 2x + x2 = 1001

Factor a perfect square on the left side:
(x + 1)(x + 1) = 1001

Calculate the square root of the right side: 31.638584039

Break this problem into two subproblems by setting 
(x + 1) equal to 31.638584039 and -31.638584039.

Subproblem 1

x + 1 = 31.638584039 Simplifying x + 1 = 31.638584039 Reorder the terms: 1 + x = 31.638584039 Solving 1 + x = 31.638584039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 31.638584039 + -1 Combine like terms: 1 + -1 = 0 0 + x = 31.638584039 + -1 x = 31.638584039 + -1 Combine like terms: 31.638584039 + -1 = 30.638584039 x = 30.638584039 Simplifying x = 30.638584039

Subproblem 2

x + 1 = -31.638584039 Simplifying x + 1 = -31.638584039 Reorder the terms: 1 + x = -31.638584039 Solving 1 + x = -31.638584039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -31.638584039 + -1 Combine like terms: 1 + -1 = 0 0 + x = -31.638584039 + -1 x = -31.638584039 + -1 Combine like terms: -31.638584039 + -1 = -32.638584039 x = -32.638584039 Simplifying x = -32.638584039

Solution

The solution to the problem is based on the solutions from the subproblems. x = {30.638584039, -32.638584039}

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