500x+0.1x^2+17526=1100x-0.01x^2

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Solution for 500x+0.1x^2+17526=1100x-0.01x^2 equation:


Simplifying
500x + 0.1x2 + 17526 = 1100x + -0.01x2

Reorder the terms:
17526 + 500x + 0.1x2 = 1100x + -0.01x2

Solving
17526 + 500x + 0.1x2 = 1100x + -0.01x2

Solving for variable 'x'.

Reorder the terms:
17526 + 500x + -1100x + 0.1x2 + 0.01x2 = 1100x + -0.01x2 + -1100x + 0.01x2

Combine like terms: 500x + -1100x = -600x
17526 + -600x + 0.1x2 + 0.01x2 = 1100x + -0.01x2 + -1100x + 0.01x2

Combine like terms: 0.1x2 + 0.01x2 = 0.11x2
17526 + -600x + 0.11x2 = 1100x + -0.01x2 + -1100x + 0.01x2

Reorder the terms:
17526 + -600x + 0.11x2 = 1100x + -1100x + -0.01x2 + 0.01x2

Combine like terms: 1100x + -1100x = 0
17526 + -600x + 0.11x2 = 0 + -0.01x2 + 0.01x2
17526 + -600x + 0.11x2 = -0.01x2 + 0.01x2

Combine like terms: -0.01x2 + 0.01x2 = 0.00
17526 + -600x + 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'.
159327.2727 + -5454.545455x + x2 = 0

Move the constant term to the right:

Add '-159327.2727' to each side of the equation.
159327.2727 + -5454.545455x + -159327.2727 + x2 = 0 + -159327.2727

Reorder the terms:
159327.2727 + -159327.2727 + -5454.545455x + x2 = 0 + -159327.2727

Combine like terms: 159327.2727 + -159327.2727 = 0.0000
0.0000 + -5454.545455x + x2 = 0 + -159327.2727
-5454.545455x + x2 = 0 + -159327.2727

Combine like terms: 0 + -159327.2727 = -159327.2727
-5454.545455x + x2 = -159327.2727

The x term is -5454.545455x.  Take half its coefficient (-2727.272728).
Square it (7438016.533) and add it to both sides.

Add '7438016.533' to each side of the equation.
-5454.545455x + 7438016.533 + x2 = -159327.2727 + 7438016.533

Reorder the terms:
7438016.533 + -5454.545455x + x2 = -159327.2727 + 7438016.533

Combine like terms: -159327.2727 + 7438016.533 = 7278689.2603
7438016.533 + -5454.545455x + x2 = 7278689.2603

Factor a perfect square on the left side:
(x + -2727.272728)(x + -2727.272728) = 7278689.2603

Calculate the square root of the right side: 2697.904605486

Break this problem into two subproblems by setting 
(x + -2727.272728) equal to 2697.904605486 and -2697.904605486.

Subproblem 1

x + -2727.272728 = 2697.904605486 Simplifying x + -2727.272728 = 2697.904605486 Reorder the terms: -2727.272728 + x = 2697.904605486 Solving -2727.272728 + x = 2697.904605486 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2727.272728' to each side of the equation. -2727.272728 + 2727.272728 + x = 2697.904605486 + 2727.272728 Combine like terms: -2727.272728 + 2727.272728 = 0.000000 0.000000 + x = 2697.904605486 + 2727.272728 x = 2697.904605486 + 2727.272728 Combine like terms: 2697.904605486 + 2727.272728 = 5425.177333486 x = 5425.177333486 Simplifying x = 5425.177333486

Subproblem 2

x + -2727.272728 = -2697.904605486 Simplifying x + -2727.272728 = -2697.904605486 Reorder the terms: -2727.272728 + x = -2697.904605486 Solving -2727.272728 + x = -2697.904605486 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2727.272728' to each side of the equation. -2727.272728 + 2727.272728 + x = -2697.904605486 + 2727.272728 Combine like terms: -2727.272728 + 2727.272728 = 0.000000 0.000000 + x = -2697.904605486 + 2727.272728 x = -2697.904605486 + 2727.272728 Combine like terms: -2697.904605486 + 2727.272728 = 29.368122514 x = 29.368122514 Simplifying x = 29.368122514

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5425.177333486, 29.368122514}

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