To find the number of rows in the auditorium when there are a total of 416 seats, we need to solve the equation:
\[ f(x) = 2x^2 - 6x = 416 \]
First, we set the equation equal to 416:
\[ 2x^2 - 6x = 416 \]
Next, we rearrange it into standard form by moving 416 to the left side:
\[ 2x^2 - 6x - 416 = 0 \]
Now, we can simplify the equation by dividing all terms by 2:
\[ x^2 - 3x - 208 = 0 \]
To solve this quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -3 \), and \( c = -208 \).
Calculating the discriminant (\( b^2 - 4ac \)):
\[ b^2 - 4ac = (-3)^2 - 4(1)(-208) = 9 + 832 = 841 \]
Now we can substitute back into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{841}}{2(1)} = \frac{3 \pm 29}{2} \]
Calculating the two possible values for \( x \):
\[ x = \frac{3 + 29}{2} = \frac{32}{2} = 16 \]
And
\[ x = \frac{3 - 29}{2} = \frac{-26}{2} = -13 \]
Since a negative number of rows doesn't make sense in this context, we discard \( x = -13\).
Thus, the number of rows in the auditorium is
\[ \boxed{16} \]