(x)=(400+20x)(90-x)

Solution for (x)=(400+20x)(90-x) equation:

Simplifying
(x) = (400 + 20x)(90 + -1x)
x = (400 + 20x)(90 + -1x)

Multiply (400 + 20x) * (90 + -1x)
x = (400(90 + -1x) + 20x * (90 + -1x))
x = ((90 * 400 + -1x * 400) + 20x * (90 + -1x))
x = ((36000 + -400x) + 20x * (90 + -1x))
x = (36000 + -400x + (90 * 20x + -1x * 20x))
x = (36000 + -400x + (1800x + -20x2))

Combine like terms: -400x + 1800x = 1400x
x = (36000 + 1400x + -20x2)

Solving
x = 36000 + 1400x + -20x2

Solving for variable 'x'.

Reorder the terms:
-36000 + x + -1400x + 20x2 = 36000 + 1400x + -20x2 + -36000 + -1400x + 20x2

Combine like terms: x + -1400x = -1399x
-36000 + -1399x + 20x2 = 36000 + 1400x + -20x2 + -36000 + -1400x + 20x2

Reorder the terms:
-36000 + -1399x + 20x2 = 36000 + -36000 + 1400x + -1400x + -20x2 + 20x2

Combine like terms: 36000 + -36000 = 0
-36000 + -1399x + 20x2 = 0 + 1400x + -1400x + -20x2 + 20x2
-36000 + -1399x + 20x2 = 1400x + -1400x + -20x2 + 20x2

Combine like terms: 1400x + -1400x = 0
-36000 + -1399x + 20x2 = 0 + -20x2 + 20x2
-36000 + -1399x + 20x2 = -20x2 + 20x2

Combine like terms: -20x2 + 20x2 = 0
-36000 + -1399x + 20x2 = 0

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

Divide each side by '20'.
-1800 + -69.95x + x2 = 0

Move the constant term to the right:

Add '1800' to each side of the equation.
-1800 + -69.95x + 1800 + x2 = 0 + 1800

Reorder the terms:
-1800 + 1800 + -69.95x + x2 = 0 + 1800

Combine like terms: -1800 + 1800 = 0
0 + -69.95x + x2 = 0 + 1800
-69.95x + x2 = 0 + 1800

Combine like terms: 0 + 1800 = 1800
-69.95x + x2 = 1800

The x term is -69.95x.  Take half its coefficient (-34.975).
Square it (1223.250625) and add it to both sides.

Add '1223.250625' to each side of the equation.
-69.95x + 1223.250625 + x2 = 1800 + 1223.250625

Reorder the terms:
1223.250625 + -69.95x + x2 = 1800 + 1223.250625

Combine like terms: 1800 + 1223.250625 = 3023.250625
1223.250625 + -69.95x + x2 = 3023.250625

Factor a perfect square on the left side:
(x + -34.975)(x + -34.975) = 3023.250625

Calculate the square root of the right side: 54.984094291

Break this problem into two subproblems by setting 
(x + -34.975) equal to 54.984094291 and -54.984094291.

Subproblem 1
x + -34.975 = 54.984094291

Simplifying
x + -34.975 = 54.984094291

Reorder the terms:
-34.975 + x = 54.984094291

Solving
-34.975 + x = 54.984094291

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '34.975' to each side of the equation.
-34.975 + 34.975 + x = 54.984094291 + 34.975

Combine like terms: -34.975 + 34.975 = 0.000
0.000 + x = 54.984094291 + 34.975
x = 54.984094291 + 34.975

Combine like terms: 54.984094291 + 34.975 = 89.959094291
x = 89.959094291

Simplifying
x = 89.959094291

Subproblem 2
x + -34.975 = -54.984094291

Simplifying
x + -34.975 = -54.984094291

Reorder the terms:
-34.975 + x = -54.984094291

Solving
-34.975 + x = -54.984094291

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '34.975' to each side of the equation.
-34.975 + 34.975 + x = -54.984094291 + 34.975

Combine like terms: -34.975 + 34.975 = 0.000
0.000 + x = -54.984094291 + 34.975
x = -54.984094291 + 34.975

Combine like terms: -54.984094291 + 34.975 = -20.009094291
x = -20.009094291

Simplifying
x = -20.009094291

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {89.959094291, -20.009094291}