x^2+20x+250=x(75-x)

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Solution for x^2+20x+250=x(75-x) equation:


Simplifying
x2 + 20x + 250 = x(75 + -1x)

Reorder the terms:
250 + 20x + x2 = x(75 + -1x)
250 + 20x + x2 = (75 * x + -1x * x)
250 + 20x + x2 = (75x + -1x2)

Solving
250 + 20x + x2 = 75x + -1x2

Solving for variable 'x'.

Reorder the terms:
250 + 20x + -75x + x2 + x2 = 75x + -1x2 + -75x + x2

Combine like terms: 20x + -75x = -55x
250 + -55x + x2 + x2 = 75x + -1x2 + -75x + x2

Combine like terms: x2 + x2 = 2x2
250 + -55x + 2x2 = 75x + -1x2 + -75x + x2

Reorder the terms:
250 + -55x + 2x2 = 75x + -75x + -1x2 + x2

Combine like terms: 75x + -75x = 0
250 + -55x + 2x2 = 0 + -1x2 + x2
250 + -55x + 2x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
250 + -55x + 2x2 = 0

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

Divide each side by '2'.
125 + -27.5x + x2 = 0

Move the constant term to the right:

Add '-125' to each side of the equation.
125 + -27.5x + -125 + x2 = 0 + -125

Reorder the terms:
125 + -125 + -27.5x + x2 = 0 + -125

Combine like terms: 125 + -125 = 0
0 + -27.5x + x2 = 0 + -125
-27.5x + x2 = 0 + -125

Combine like terms: 0 + -125 = -125
-27.5x + x2 = -125

The x term is -27.5x.  Take half its coefficient (-13.75).
Square it (189.0625) and add it to both sides.

Add '189.0625' to each side of the equation.
-27.5x + 189.0625 + x2 = -125 + 189.0625

Reorder the terms:
189.0625 + -27.5x + x2 = -125 + 189.0625

Combine like terms: -125 + 189.0625 = 64.0625
189.0625 + -27.5x + x2 = 64.0625

Factor a perfect square on the left side:
(x + -13.75)(x + -13.75) = 64.0625

Calculate the square root of the right side: 8.003905297

Break this problem into two subproblems by setting 
(x + -13.75) equal to 8.003905297 and -8.003905297.

Subproblem 1

x + -13.75 = 8.003905297 Simplifying x + -13.75 = 8.003905297 Reorder the terms: -13.75 + x = 8.003905297 Solving -13.75 + x = 8.003905297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.75' to each side of the equation. -13.75 + 13.75 + x = 8.003905297 + 13.75 Combine like terms: -13.75 + 13.75 = 0.00 0.00 + x = 8.003905297 + 13.75 x = 8.003905297 + 13.75 Combine like terms: 8.003905297 + 13.75 = 21.753905297 x = 21.753905297 Simplifying x = 21.753905297

Subproblem 2

x + -13.75 = -8.003905297 Simplifying x + -13.75 = -8.003905297 Reorder the terms: -13.75 + x = -8.003905297 Solving -13.75 + x = -8.003905297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.75' to each side of the equation. -13.75 + 13.75 + x = -8.003905297 + 13.75 Combine like terms: -13.75 + 13.75 = 0.00 0.00 + x = -8.003905297 + 13.75 x = -8.003905297 + 13.75 Combine like terms: -8.003905297 + 13.75 = 5.746094703 x = 5.746094703 Simplifying x = 5.746094703

Solution

The solution to the problem is based on the solutions from the subproblems. x = {21.753905297, 5.746094703}

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