100x(x-3)=450-50x

Solution for 100x(x-3)=450-50x equation:

Simplifying
100x(x + -3) = 450 + -50x

Reorder the terms:
100x(-3 + x) = 450 + -50x
(-3 * 100x + x * 100x) = 450 + -50x
(-300x + 100x2) = 450 + -50x

Solving
-300x + 100x2 = 450 + -50x

Solving for variable 'x'.

Reorder the terms:
-450 + -300x + 50x + 100x2 = 450 + -50x + -450 + 50x

Combine like terms: -300x + 50x = -250x
-450 + -250x + 100x2 = 450 + -50x + -450 + 50x

Reorder the terms:
-450 + -250x + 100x2 = 450 + -450 + -50x + 50x

Combine like terms: 450 + -450 = 0
-450 + -250x + 100x2 = 0 + -50x + 50x
-450 + -250x + 100x2 = -50x + 50x

Combine like terms: -50x + 50x = 0
-450 + -250x + 100x2 = 0

Factor out the Greatest Common Factor (GCF), '50'.
50(-9 + -5x + 2x2) = 0

Ignore the factor 50.
Subproblem 1
Set the factor '(-9 + -5x + 2x2)' equal to zero and attempt to solve:

Simplifying
-9 + -5x + 2x2 = 0

Solving
-9 + -5x + 2x2 = 0

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

Divide each side by '2'.
-4.5 + -2.5x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -2.5x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -2.5x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -2.5x + x2 = 0 + 4.5
-2.5x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-2.5x + x2 = 4.5

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 4.5 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 4.5 + 1.5625

Combine like terms: 4.5 + 1.5625 = 6.0625
1.5625 + -2.5x + x2 = 6.0625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(x + -1.25) equal to 2.46221445 and -2.46221445.

Subproblem 1
x + -1.25 = 2.46221445

Simplifying
x + -1.25 = 2.46221445

Reorder the terms:
-1.25 + x = 2.46221445

Solving
-1.25 + x = 2.46221445

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = 2.46221445 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = 2.46221445 + 1.25
x = 2.46221445 + 1.25

Combine like terms: 2.46221445 + 1.25 = 3.71221445
x = 3.71221445

Simplifying
x = 3.71221445

Subproblem 2
x + -1.25 = -2.46221445

Simplifying
x + -1.25 = -2.46221445

Reorder the terms:
-1.25 + x = -2.46221445

Solving
-1.25 + x = -2.46221445

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = -2.46221445 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = -2.46221445 + 1.25
x = -2.46221445 + 1.25

Combine like terms: -2.46221445 + 1.25 = -1.21221445
x = -1.21221445

Simplifying
x = -1.21221445

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.71221445, -1.21221445}
Solution
x = {3.71221445, -1.21221445}