x^2+x=1500

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Solution for x^2+x=1500 equation:


Simplifying
x2 + x = 1500

Reorder the terms:
x + x2 = 1500

Solving
x + x2 = 1500

Solving for variable 'x'.

Reorder the terms:
-1500 + x + x2 = 1500 + -1500

Combine like terms: 1500 + -1500 = 0
-1500 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1500' to each side of the equation.
-1500 + x + 1500 + x2 = 0 + 1500

Reorder the terms:
-1500 + 1500 + x + x2 = 0 + 1500

Combine like terms: -1500 + 1500 = 0
0 + x + x2 = 0 + 1500
x + x2 = 0 + 1500

Combine like terms: 0 + 1500 = 1500
x + x2 = 1500

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 = 1500 + 0.25

Reorder the terms:
0.25 + x + x2 = 1500 + 0.25

Combine like terms: 1500 + 0.25 = 1500.25
0.25 + x + x2 = 1500.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1500.25

Calculate the square root of the right side: 38.733060814

Break this problem into two subproblems by setting 
(x + 0.5) equal to 38.733060814 and -38.733060814.

Subproblem 1

x + 0.5 = 38.733060814 Simplifying x + 0.5 = 38.733060814 Reorder the terms: 0.5 + x = 38.733060814 Solving 0.5 + x = 38.733060814 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 = 38.733060814 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 38.733060814 + -0.5 x = 38.733060814 + -0.5 Combine like terms: 38.733060814 + -0.5 = 38.233060814 x = 38.233060814 Simplifying x = 38.233060814

Subproblem 2

x + 0.5 = -38.733060814 Simplifying x + 0.5 = -38.733060814 Reorder the terms: 0.5 + x = -38.733060814 Solving 0.5 + x = -38.733060814 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 = -38.733060814 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -38.733060814 + -0.5 x = -38.733060814 + -0.5 Combine like terms: -38.733060814 + -0.5 = -39.233060814 x = -39.233060814 Simplifying x = -39.233060814

Solution

The solution to the problem is based on the solutions from the subproblems. x = {38.233060814, -39.233060814}

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