Question

The total 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 n the auditorium if it has a total of 416 seats?

Answer 1

To find the number of rows \( x \) in the auditorium with a total of 416 seats, we need to set the function \( f(x) = 2x^2 - 6x \) equal to 416 and solve for \( x \):

\[
f(x) = 2x^2 - 6x = 416
\]

First, we will set the equation:

\[
2x^2 - 6x - 416 = 0
\]

Next, we can simplify this equation by dividing all terms by 2:

\[
x^2 - 3x - 208 = 0
\]

Now, we will use the quadratic formula to find the roots of this equation. The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our equation, \( a = 1 \), \( b = -3 \), and \( c = -208 \). Plugging in these values:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-208)}}{2 \cdot 1}
\]

Calculating inside the square root:

\[
x = \frac{3 \pm \sqrt{9 + 832}}{2}
\]
\[
x = \frac{3 \pm \sqrt{841}}{2}
\]
\[
x = \frac{3 \pm 29}{2}
\]

Now we solve for the two possible values of \( x \):

1.

\[
x = \frac{3 + 29}{2} = \frac{32}{2} = 16
\]

2.

\[
x = \frac{3 - 29}{2} = \frac{-26}{2} = -13
\]

Since \( x \) represents the number of rows of seats, it must be a non-negative value. Therefore, we discard \( x = -13 \) and keep:

\[
x = 16
\]

Thus, the auditorium has \(\boxed{16}\) rows of seats.