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 \( x \):

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

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

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.