x^2=x+1000

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Solution for x^2=x+1000 equation:


Simplifying
x2 = x + 1000

Reorder the terms:
x2 = 1000 + x

Solving
x2 = 1000 + x

Solving for variable 'x'.

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

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

Combine like terms: 1000 + -1000 = 0
-1000 + -1x + x2 = 0 + x + -1x
-1000 + -1x + x2 = x + -1x

Combine like terms: x + -1x = 0
-1000 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1000' to each side of the equation.
-1000 + -1x + 1000 + x2 = 0 + 1000

Reorder the terms:
-1000 + 1000 + -1x + x2 = 0 + 1000

Combine like terms: -1000 + 1000 = 0
0 + -1x + x2 = 0 + 1000
-1x + x2 = 0 + 1000

Combine like terms: 0 + 1000 = 1000
-1x + x2 = 1000

The x term is -1x.  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.
-1x + 0.25 + x2 = 1000 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 1000 + 0.25

Combine like terms: 1000 + 0.25 = 1000.25
0.25 + -1x + x2 = 1000.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 1000.25

Calculate the square root of the right side: 31.626729202

Break this problem into two subproblems by setting 
(x + -0.5) equal to 31.626729202 and -31.626729202.

Subproblem 1

x + -0.5 = 31.626729202 Simplifying x + -0.5 = 31.626729202 Reorder the terms: -0.5 + x = 31.626729202 Solving -0.5 + x = 31.626729202 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 = 31.626729202 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 31.626729202 + 0.5 x = 31.626729202 + 0.5 Combine like terms: 31.626729202 + 0.5 = 32.126729202 x = 32.126729202 Simplifying x = 32.126729202

Subproblem 2

x + -0.5 = -31.626729202 Simplifying x + -0.5 = -31.626729202 Reorder the terms: -0.5 + x = -31.626729202 Solving -0.5 + x = -31.626729202 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 = -31.626729202 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -31.626729202 + 0.5 x = -31.626729202 + 0.5 Combine like terms: -31.626729202 + 0.5 = -31.126729202 x = -31.126729202 Simplifying x = -31.126729202

Solution

The solution to the problem is based on the solutions from the subproblems. x = {32.126729202, -31.126729202}

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