500x+0.1x^2+17764=1000x-0.01x^2

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Solution for 500x+0.1x^2+17764=1000x-0.01x^2 equation:


Simplifying
500x + 0.1x2 + 17764 = 1000x + -0.01x2

Reorder the terms:
17764 + 500x + 0.1x2 = 1000x + -0.01x2

Solving
17764 + 500x + 0.1x2 = 1000x + -0.01x2

Solving for variable 'x'.

Reorder the terms:
17764 + 500x + -1000x + 0.1x2 + 0.01x2 = 1000x + -0.01x2 + -1000x + 0.01x2

Combine like terms: 500x + -1000x = -500x
17764 + -500x + 0.1x2 + 0.01x2 = 1000x + -0.01x2 + -1000x + 0.01x2

Combine like terms: 0.1x2 + 0.01x2 = 0.11x2
17764 + -500x + 0.11x2 = 1000x + -0.01x2 + -1000x + 0.01x2

Reorder the terms:
17764 + -500x + 0.11x2 = 1000x + -1000x + -0.01x2 + 0.01x2

Combine like terms: 1000x + -1000x = 0
17764 + -500x + 0.11x2 = 0 + -0.01x2 + 0.01x2
17764 + -500x + 0.11x2 = -0.01x2 + 0.01x2

Combine like terms: -0.01x2 + 0.01x2 = 0.00
17764 + -500x + 0.11x2 = 0.00

Begin completing the square.  Divide all terms by
0.11 the coefficient of the squared term: 

Divide each side by '0.11'.
161490.9091 + -4545.454545x + x2 = 0

Move the constant term to the right:

Add '-161490.9091' to each side of the equation.
161490.9091 + -4545.454545x + -161490.9091 + x2 = 0 + -161490.9091

Reorder the terms:
161490.9091 + -161490.9091 + -4545.454545x + x2 = 0 + -161490.9091

Combine like terms: 161490.9091 + -161490.9091 = 0.0000
0.0000 + -4545.454545x + x2 = 0 + -161490.9091
-4545.454545x + x2 = 0 + -161490.9091

Combine like terms: 0 + -161490.9091 = -161490.9091
-4545.454545x + x2 = -161490.9091

The x term is -4545.454545x.  Take half its coefficient (-2272.727273).
Square it (5165289.257) and add it to both sides.

Add '5165289.257' to each side of the equation.
-4545.454545x + 5165289.257 + x2 = -161490.9091 + 5165289.257

Reorder the terms:
5165289.257 + -4545.454545x + x2 = -161490.9091 + 5165289.257

Combine like terms: -161490.9091 + 5165289.257 = 5003798.3479
5165289.257 + -4545.454545x + x2 = 5003798.3479

Factor a perfect square on the left side:
(x + -2272.727273)(x + -2272.727273) = 5003798.3479

Calculate the square root of the right side: 2236.917152668

Break this problem into two subproblems by setting 
(x + -2272.727273) equal to 2236.917152668 and -2236.917152668.

Subproblem 1

x + -2272.727273 = 2236.917152668 Simplifying x + -2272.727273 = 2236.917152668 Reorder the terms: -2272.727273 + x = 2236.917152668 Solving -2272.727273 + x = 2236.917152668 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2272.727273' to each side of the equation. -2272.727273 + 2272.727273 + x = 2236.917152668 + 2272.727273 Combine like terms: -2272.727273 + 2272.727273 = 0.000000 0.000000 + x = 2236.917152668 + 2272.727273 x = 2236.917152668 + 2272.727273 Combine like terms: 2236.917152668 + 2272.727273 = 4509.644425668 x = 4509.644425668 Simplifying x = 4509.644425668

Subproblem 2

x + -2272.727273 = -2236.917152668 Simplifying x + -2272.727273 = -2236.917152668 Reorder the terms: -2272.727273 + x = -2236.917152668 Solving -2272.727273 + x = -2236.917152668 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2272.727273' to each side of the equation. -2272.727273 + 2272.727273 + x = -2236.917152668 + 2272.727273 Combine like terms: -2272.727273 + 2272.727273 = 0.000000 0.000000 + x = -2236.917152668 + 2272.727273 x = -2236.917152668 + 2272.727273 Combine like terms: -2236.917152668 + 2272.727273 = 35.810120332 x = 35.810120332 Simplifying x = 35.810120332

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4509.644425668, 35.810120332}

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