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?

32
13
20
16

Answer 1

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

\[
\boxed{16}
\]