11x(19-x)+13x=209

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Solution for 11x(19-x)+13x=209 equation:


Simplifying
11x(19 + -1x) + 13x = 209
(19 * 11x + -1x * 11x) + 13x = 209
(209x + -11x2) + 13x = 209

Reorder the terms:
209x + 13x + -11x2 = 209

Combine like terms: 209x + 13x = 222x
222x + -11x2 = 209

Solving
222x + -11x2 = 209

Solving for variable 'x'.

Reorder the terms:
-209 + 222x + -11x2 = 209 + -209

Combine like terms: 209 + -209 = 0
-209 + 222x + -11x2 = 0

Begin completing the square.  Divide all terms by
-11 the coefficient of the squared term: 

Divide each side by '-11'.
19 + -20.18181818x + x2 = 0

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + -20.18181818x + -19 + x2 = 0 + -19

Reorder the terms:
19 + -19 + -20.18181818x + x2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + -20.18181818x + x2 = 0 + -19
-20.18181818x + x2 = 0 + -19

Combine like terms: 0 + -19 = -19
-20.18181818x + x2 = -19

The x term is -20.18181818x.  Take half its coefficient (-10.09090909).
Square it (101.8264463) and add it to both sides.

Add '101.8264463' to each side of the equation.
-20.18181818x + 101.8264463 + x2 = -19 + 101.8264463

Reorder the terms:
101.8264463 + -20.18181818x + x2 = -19 + 101.8264463

Combine like terms: -19 + 101.8264463 = 82.8264463
101.8264463 + -20.18181818x + x2 = 82.8264463

Factor a perfect square on the left side:
(x + -10.09090909)(x + -10.09090909) = 82.8264463

Calculate the square root of the right side: 9.100903598

Break this problem into two subproblems by setting 
(x + -10.09090909) equal to 9.100903598 and -9.100903598.

Subproblem 1

x + -10.09090909 = 9.100903598 Simplifying x + -10.09090909 = 9.100903598 Reorder the terms: -10.09090909 + x = 9.100903598 Solving -10.09090909 + x = 9.100903598 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.09090909' to each side of the equation. -10.09090909 + 10.09090909 + x = 9.100903598 + 10.09090909 Combine like terms: -10.09090909 + 10.09090909 = 0.00000000 0.00000000 + x = 9.100903598 + 10.09090909 x = 9.100903598 + 10.09090909 Combine like terms: 9.100903598 + 10.09090909 = 19.191812688 x = 19.191812688 Simplifying x = 19.191812688

Subproblem 2

x + -10.09090909 = -9.100903598 Simplifying x + -10.09090909 = -9.100903598 Reorder the terms: -10.09090909 + x = -9.100903598 Solving -10.09090909 + x = -9.100903598 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.09090909' to each side of the equation. -10.09090909 + 10.09090909 + x = -9.100903598 + 10.09090909 Combine like terms: -10.09090909 + 10.09090909 = 0.00000000 0.00000000 + x = -9.100903598 + 10.09090909 x = -9.100903598 + 10.09090909 Combine like terms: -9.100903598 + 10.09090909 = 0.990005492 x = 0.990005492 Simplifying x = 0.990005492

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.191812688, 0.990005492}

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