x^2+x=1007.5

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Solution for x^2+x=1007.5 equation: 


Simplifying
x2 + x = 1007.5

Reorder the terms:
x + x2 = 1007.5

Solving
x + x2 = 1007.5

Solving for variable 'x'.

Reorder the terms:
-1007.5 + x + x2 = 1007.5 + -1007.5

Combine like terms: 1007.5 + -1007.5 = 0.0
-1007.5 + x + x2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '1007.5' to each side of the equation.
-1007.5 + x + 1007.5 + x2 = 0.0 + 1007.5

Reorder the terms:
-1007.5 + 1007.5 + x + x2 = 0.0 + 1007.5

Combine like terms: -1007.5 + 1007.5 = 0.0
0.0 + x + x2 = 0.0 + 1007.5
x + x2 = 0.0 + 1007.5

Combine like terms: 0.0 + 1007.5 = 1007.5
x + x2 = 1007.5

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 = 1007.5 + 0.25

Reorder the terms:
0.25 + x + x2 = 1007.5 + 0.25

Combine like terms: 1007.5 + 0.25 = 1007.75
0.25 + x + x2 = 1007.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1007.75

Calculate the square root of the right side: 31.745078359

Break this problem into two subproblems by setting 
(x + 0.5) equal to 31.745078359 and -31.745078359.


Subproblem 1
x + 0.5 = 31.745078359

Simplifying
x + 0.5 = 31.745078359

Reorder the terms:
0.5 + x = 31.745078359

Solving
0.5 + x = 31.745078359

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 = 31.745078359 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 31.745078359 + -0.5
x = 31.745078359 + -0.5

Combine like terms: 31.745078359 + -0.5 = 31.245078359
x = 31.245078359

Simplifying
x = 31.245078359


Subproblem 2
x + 0.5 = -31.745078359

Simplifying
x + 0.5 = -31.745078359

Reorder the terms:
0.5 + x = -31.745078359

Solving
0.5 + x = -31.745078359

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 = -31.745078359 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -31.745078359 + -0.5
x = -31.745078359 + -0.5

Combine like terms: -31.745078359 + -0.5 = -32.245078359
x = -32.245078359

Simplifying
x = -32.245078359


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {31.245078359, -32.245078359}

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