54600=(550+25n)(96+3n)

Solution for 54600=(550+25n)(96+3n) equation:

Simplifying
54600 = (550 + 25n)(96 + 3n)

Multiply (550 + 25n) * (96 + 3n)
54600 = (550(96 + 3n) + 25n * (96 + 3n))
54600 = ((96 * 550 + 3n * 550) + 25n * (96 + 3n))
54600 = ((52800 + 1650n) + 25n * (96 + 3n))
54600 = (52800 + 1650n + (96 * 25n + 3n * 25n))
54600 = (52800 + 1650n + (2400n + 75n2))

Combine like terms: 1650n + 2400n = 4050n
54600 = (52800 + 4050n + 75n2)

Solving
54600 = 52800 + 4050n + 75n2

Solving for variable 'n'.

Combine like terms: 54600 + -52800 = 1800
1800 + -4050n + -75n2 = 52800 + 4050n + 75n2 + -52800 + -4050n + -75n2

Reorder the terms:
1800 + -4050n + -75n2 = 52800 + -52800 + 4050n + -4050n + 75n2 + -75n2

Combine like terms: 52800 + -52800 = 0
1800 + -4050n + -75n2 = 0 + 4050n + -4050n + 75n2 + -75n2
1800 + -4050n + -75n2 = 4050n + -4050n + 75n2 + -75n2

Combine like terms: 4050n + -4050n = 0
1800 + -4050n + -75n2 = 0 + 75n2 + -75n2
1800 + -4050n + -75n2 = 75n2 + -75n2

Combine like terms: 75n2 + -75n2 = 0
1800 + -4050n + -75n2 = 0

Factor out the Greatest Common Factor (GCF), '75'.
75(24 + -54n + -1n2) = 0

Ignore the factor 75.
Subproblem 1
Set the factor '(24 + -54n + -1n2)' equal to zero and attempt to solve:

Simplifying
24 + -54n + -1n2 = 0

Solving
24 + -54n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-24 + 54n + n2 = 0

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + 54n + 24 + n2 = 0 + 24

Reorder the terms:
-24 + 24 + 54n + n2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + 54n + n2 = 0 + 24
54n + n2 = 0 + 24

Combine like terms: 0 + 24 = 24
54n + n2 = 24

The n term is 54n.  Take half its coefficient (27).
Square it (729) and add it to both sides.

Add '729' to each side of the equation.
54n + 729 + n2 = 24 + 729

Reorder the terms:
729 + 54n + n2 = 24 + 729

Combine like terms: 24 + 729 = 753
729 + 54n + n2 = 753

Factor a perfect square on the left side:
(n + 27)(n + 27) = 753

Calculate the square root of the right side: 27.440845468

Break this problem into two subproblems by setting 
(n + 27) equal to 27.440845468 and -27.440845468.

Subproblem 1
n + 27 = 27.440845468

Simplifying
n + 27 = 27.440845468

Reorder the terms:
27 + n = 27.440845468

Solving
27 + n = 27.440845468

Solving for variable 'n'.

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

Add '-27' to each side of the equation.
27 + -27 + n = 27.440845468 + -27

Combine like terms: 27 + -27 = 0
0 + n = 27.440845468 + -27
n = 27.440845468 + -27

Combine like terms: 27.440845468 + -27 = 0.440845468
n = 0.440845468

Simplifying
n = 0.440845468

Subproblem 2
n + 27 = -27.440845468

Simplifying
n + 27 = -27.440845468

Reorder the terms:
27 + n = -27.440845468

Solving
27 + n = -27.440845468

Solving for variable 'n'.

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

Add '-27' to each side of the equation.
27 + -27 + n = -27.440845468 + -27

Combine like terms: 27 + -27 = 0
0 + n = -27.440845468 + -27
n = -27.440845468 + -27

Combine like terms: -27.440845468 + -27 = -54.440845468
n = -54.440845468

Simplifying
n = -54.440845468

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.440845468, -54.440845468}
Solution
n = {0.440845468, -54.440845468}