20+5x-0.1x^2=10x-0.25x^2

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Solution for 20+5x-0.1x^2=10x-0.25x^2 equation:


Simplifying
20 + 5x + -0.1x2 = 10x + -0.25x2

Solving
20 + 5x + -0.1x2 = 10x + -0.25x2

Solving for variable 'x'.

Reorder the terms:
20 + 5x + -10x + -0.1x2 + 0.25x2 = 10x + -0.25x2 + -10x + 0.25x2

Combine like terms: 5x + -10x = -5x
20 + -5x + -0.1x2 + 0.25x2 = 10x + -0.25x2 + -10x + 0.25x2

Combine like terms: -0.1x2 + 0.25x2 = 0.15x2
20 + -5x + 0.15x2 = 10x + -0.25x2 + -10x + 0.25x2

Reorder the terms:
20 + -5x + 0.15x2 = 10x + -10x + -0.25x2 + 0.25x2

Combine like terms: 10x + -10x = 0
20 + -5x + 0.15x2 = 0 + -0.25x2 + 0.25x2
20 + -5x + 0.15x2 = -0.25x2 + 0.25x2

Combine like terms: -0.25x2 + 0.25x2 = 0.00
20 + -5x + 0.15x2 = 0.00

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

Divide each side by '0.15'.
133.3333333 + -33.33333333x + x2 = 0

Move the constant term to the right:

Add '-133.3333333' to each side of the equation.
133.3333333 + -33.33333333x + -133.3333333 + x2 = 0 + -133.3333333

Reorder the terms:
133.3333333 + -133.3333333 + -33.33333333x + x2 = 0 + -133.3333333

Combine like terms: 133.3333333 + -133.3333333 = 0.0000000
0.0000000 + -33.33333333x + x2 = 0 + -133.3333333
-33.33333333x + x2 = 0 + -133.3333333

Combine like terms: 0 + -133.3333333 = -133.3333333
-33.33333333x + x2 = -133.3333333

The x term is -33.33333333x.  Take half its coefficient (-16.66666667).
Square it (277.7777779) and add it to both sides.

Add '277.7777779' to each side of the equation.
-33.33333333x + 277.7777779 + x2 = -133.3333333 + 277.7777779

Reorder the terms:
277.7777779 + -33.33333333x + x2 = -133.3333333 + 277.7777779

Combine like terms: -133.3333333 + 277.7777779 = 144.4444446
277.7777779 + -33.33333333x + x2 = 144.4444446

Factor a perfect square on the left side:
(x + -16.66666667)(x + -16.66666667) = 144.4444446

Calculate the square root of the right side: 12.018504258

Break this problem into two subproblems by setting 
(x + -16.66666667) equal to 12.018504258 and -12.018504258.

Subproblem 1

x + -16.66666667 = 12.018504258 Simplifying x + -16.66666667 = 12.018504258 Reorder the terms: -16.66666667 + x = 12.018504258 Solving -16.66666667 + x = 12.018504258 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16.66666667' to each side of the equation. -16.66666667 + 16.66666667 + x = 12.018504258 + 16.66666667 Combine like terms: -16.66666667 + 16.66666667 = 0.00000000 0.00000000 + x = 12.018504258 + 16.66666667 x = 12.018504258 + 16.66666667 Combine like terms: 12.018504258 + 16.66666667 = 28.685170928 x = 28.685170928 Simplifying x = 28.685170928

Subproblem 2

x + -16.66666667 = -12.018504258 Simplifying x + -16.66666667 = -12.018504258 Reorder the terms: -16.66666667 + x = -12.018504258 Solving -16.66666667 + x = -12.018504258 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16.66666667' to each side of the equation. -16.66666667 + 16.66666667 + x = -12.018504258 + 16.66666667 Combine like terms: -16.66666667 + 16.66666667 = 0.00000000 0.00000000 + x = -12.018504258 + 16.66666667 x = -12.018504258 + 16.66666667 Combine like terms: -12.018504258 + 16.66666667 = 4.648162412 x = 4.648162412 Simplifying x = 4.648162412

Solution

The solution to the problem is based on the solutions from the subproblems. x = {28.685170928, 4.648162412}

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