400x+0.1x^2+17686=1100x-0.01x^2

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Solution for 400x+0.1x^2+17686=1100x-0.01x^2 equation:


Simplifying
400x + 0.1x2 + 17686 = 1100x + -0.01x2

Reorder the terms:
17686 + 400x + 0.1x2 = 1100x + -0.01x2

Solving
17686 + 400x + 0.1x2 = 1100x + -0.01x2

Solving for variable 'x'.

Reorder the terms:
17686 + 400x + -1100x + 0.1x2 + 0.01x2 = 1100x + -0.01x2 + -1100x + 0.01x2

Combine like terms: 400x + -1100x = -700x
17686 + -700x + 0.1x2 + 0.01x2 = 1100x + -0.01x2 + -1100x + 0.01x2

Combine like terms: 0.1x2 + 0.01x2 = 0.11x2
17686 + -700x + 0.11x2 = 1100x + -0.01x2 + -1100x + 0.01x2

Reorder the terms:
17686 + -700x + 0.11x2 = 1100x + -1100x + -0.01x2 + 0.01x2

Combine like terms: 1100x + -1100x = 0
17686 + -700x + 0.11x2 = 0 + -0.01x2 + 0.01x2
17686 + -700x + 0.11x2 = -0.01x2 + 0.01x2

Combine like terms: -0.01x2 + 0.01x2 = 0.00
17686 + -700x + 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'.
160781.8182 + -6363.636364x + x2 = 0

Move the constant term to the right:

Add '-160781.8182' to each side of the equation.
160781.8182 + -6363.636364x + -160781.8182 + x2 = 0 + -160781.8182

Reorder the terms:
160781.8182 + -160781.8182 + -6363.636364x + x2 = 0 + -160781.8182

Combine like terms: 160781.8182 + -160781.8182 = 0.0000
0.0000 + -6363.636364x + x2 = 0 + -160781.8182
-6363.636364x + x2 = 0 + -160781.8182

Combine like terms: 0 + -160781.8182 = -160781.8182
-6363.636364x + x2 = -160781.8182

The x term is -6363.636364x.  Take half its coefficient (-3181.818182).
Square it (10123966.94) and add it to both sides.

Add '10123966.94' to each side of the equation.
-6363.636364x + 10123966.94 + x2 = -160781.8182 + 10123966.94

Reorder the terms:
10123966.94 + -6363.636364x + x2 = -160781.8182 + 10123966.94

Combine like terms: -160781.8182 + 10123966.94 = 9963185.1218
10123966.94 + -6363.636364x + x2 = 9963185.1218

Factor a perfect square on the left side:
(x + -3181.818182)(x + -3181.818182) = 9963185.1218

Calculate the square root of the right side: 3156.451349506

Break this problem into two subproblems by setting 
(x + -3181.818182) equal to 3156.451349506 and -3156.451349506.

Subproblem 1

x + -3181.818182 = 3156.451349506 Simplifying x + -3181.818182 = 3156.451349506 Reorder the terms: -3181.818182 + x = 3156.451349506 Solving -3181.818182 + x = 3156.451349506 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3181.818182' to each side of the equation. -3181.818182 + 3181.818182 + x = 3156.451349506 + 3181.818182 Combine like terms: -3181.818182 + 3181.818182 = 0.000000 0.000000 + x = 3156.451349506 + 3181.818182 x = 3156.451349506 + 3181.818182 Combine like terms: 3156.451349506 + 3181.818182 = 6338.269531506 x = 6338.269531506 Simplifying x = 6338.269531506

Subproblem 2

x + -3181.818182 = -3156.451349506 Simplifying x + -3181.818182 = -3156.451349506 Reorder the terms: -3181.818182 + x = -3156.451349506 Solving -3181.818182 + x = -3156.451349506 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3181.818182' to each side of the equation. -3181.818182 + 3181.818182 + x = -3156.451349506 + 3181.818182 Combine like terms: -3181.818182 + 3181.818182 = 0.000000 0.000000 + x = -3156.451349506 + 3181.818182 x = -3156.451349506 + 3181.818182 Combine like terms: -3156.451349506 + 3181.818182 = 25.366832494 x = 25.366832494 Simplifying x = 25.366832494

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6338.269531506, 25.366832494}

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