The number of seats in an auditorium is modeled by `f\left(x\right)=2x^{2}-6x,`where x represents the number of rows of seats. How many rows are there in the auditorium if it has a total of 416 seats?

Answer 1

To find the number of rows in the auditorium if it has a total of 416 seats, we need to solve the equation `2x^{2}-6x=416` for x.

`2x^{2}-6x=416`

Rearranging the equation, we get:

`2x^{2}-6x-416=0`

Now, we need to factor the quadratic equation:

`2x^{2}-6x-416=0`

`2(x^{2}-3x-208)=0`

`2(x-16)(x+13)=0`

Setting each factor to zero gives us:

`x-16=0` or `x+13=0`

`x=16` or `x=-13`

Since the number of rows (x) cannot be negative, the auditorium has 16 rows of seats.