The total number of seats in an auditorium is modeled by f(x) = 2x² - 12x, where x represents the number of rows of seats. how many rows are there in the auditorium if it has a total of 320 seats? show work and explain how you got your answer

320 = 2x² - 12x as when there are an unknown number of rows there are 320 seats.
0 = 2x² - 12x - 320
0 = x² - 6x - 160
0 = (x - 16)(x + 10)
for this to = 0 one bracket must equal 0
therefore x = 16 or x = -10
not a negative number of rows.

Therefore must be 16.

There are 240 ways to arrange themselves in line

Step-by-step explanation:

The number of ways that satisfied Frankie's requirement is defined by the position of Joey. Then we have the following cases:

1. Joey is the first in line: the number of ways for this case is calculated using a rule of multiplication as:

       1         *        4      *     4     *     3      *        2     *   1         =  96

1st in Line          2nd          3rd         4th          5th           6th

Because, Joey is the first in line, then Frankie can not be right behind him, so we have 4 mobsters for be 2nd in line, and 4 mobsters for be 3rd in line, including Frankie, 3 mobsters for be 4th in Line, 2 mobsters for be 5th in line and 1 mobster for be 6th in line.

2. Joey is the second in line:   the number of ways for this case is calculated using a rule of multiplication as:

       4         *        1      *     3     *     3      *        2     *   1         =  72

1st in Line          2nd          3rd         4th          5th           6th

Because Joey is the second in line and Frankie can not be in neither the 1st or 3rd position of the line.

At the same way we can calculated the number of ways for the following cases:

3. Joey is the Third in line:  

       4         *        3      *     1     *     2      *        2     *   1         =  48

1st in Line          2nd          3rd         4th          5th           6th

4. Joey is the fourth in line:  

       4         *        3      *     2     *     1      *        1     *   1         =  24

1st in Line          2nd          3rd         4th          5th           6th

On the other hand, if Joey is 5th or 6th in line, the requirement of Frankie is not satisfy. Then, the total number of ways in which Frankie is satisfied is calculated as the sum of the cases mentioned above. This is:

96 + 72 + 48 + 24 = 240 ways

So, there are 240 ways to arrange themselves in line