150x+250x(40-x)=8400

Solution for 150x+250x(40-x)=8400 equation:

Simplifying
150x + 250x(40 + -1x) = 8400
150x + (40 * 250x + -1x * 250x) = 8400
150x + (10000x + -250x2) = 8400

Combine like terms: 150x + 10000x = 10150x
10150x + -250x2 = 8400

Solving
10150x + -250x2 = 8400

Solving for variable 'x'.

Reorder the terms:
-8400 + 10150x + -250x2 = 8400 + -8400

Combine like terms: 8400 + -8400 = 0
-8400 + 10150x + -250x2 = 0

Factor out the Greatest Common Factor (GCF), '50'.
50(-168 + 203x + -5x2) = 0

Ignore the factor 50.
Subproblem 1
Set the factor '(-168 + 203x + -5x2)' equal to zero and attempt to solve:

Simplifying
-168 + 203x + -5x2 = 0

Solving
-168 + 203x + -5x2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
33.6 + -40.6x + x2 = 0

Move the constant term to the right:

Add '-33.6' to each side of the equation.
33.6 + -40.6x + -33.6 + x2 = 0 + -33.6

Reorder the terms:
33.6 + -33.6 + -40.6x + x2 = 0 + -33.6

Combine like terms: 33.6 + -33.6 = 0.0
0.0 + -40.6x + x2 = 0 + -33.6
-40.6x + x2 = 0 + -33.6

Combine like terms: 0 + -33.6 = -33.6
-40.6x + x2 = -33.6

The x term is -40.6x.  Take half its coefficient (-20.3).
Square it (412.09) and add it to both sides.

Add '412.09' to each side of the equation.
-40.6x + 412.09 + x2 = -33.6 + 412.09

Reorder the terms:
412.09 + -40.6x + x2 = -33.6 + 412.09

Combine like terms: -33.6 + 412.09 = 378.49
412.09 + -40.6x + x2 = 378.49

Factor a perfect square on the left side:
(x + -20.3)(x + -20.3) = 378.49

Calculate the square root of the right side: 19.454819454

Break this problem into two subproblems by setting 
(x + -20.3) equal to 19.454819454 and -19.454819454.

Subproblem 1
x + -20.3 = 19.454819454

Simplifying
x + -20.3 = 19.454819454

Reorder the terms:
-20.3 + x = 19.454819454

Solving
-20.3 + x = 19.454819454

Solving for variable 'x'.

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

Add '20.3' to each side of the equation.
-20.3 + 20.3 + x = 19.454819454 + 20.3

Combine like terms: -20.3 + 20.3 = 0.0
0.0 + x = 19.454819454 + 20.3
x = 19.454819454 + 20.3

Combine like terms: 19.454819454 + 20.3 = 39.754819454
x = 39.754819454

Simplifying
x = 39.754819454

Subproblem 2
x + -20.3 = -19.454819454

Simplifying
x + -20.3 = -19.454819454

Reorder the terms:
-20.3 + x = -19.454819454

Solving
-20.3 + x = -19.454819454

Solving for variable 'x'.

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

Add '20.3' to each side of the equation.
-20.3 + 20.3 + x = -19.454819454 + 20.3

Combine like terms: -20.3 + 20.3 = 0.0
0.0 + x = -19.454819454 + 20.3
x = -19.454819454 + 20.3

Combine like terms: -19.454819454 + 20.3 = 0.845180546
x = 0.845180546

Simplifying
x = 0.845180546

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {39.754819454, 0.845180546}
Solution
x = {39.754819454, 0.845180546}