A rectangular auditorium seats 1920 people. The number of seats in each row exceeds the number of rows by 8. Find the number of seats in each row.
number of rows --- x
number of seats per row --- x+8
x(x+8) = 1920
x^2 + 8x - 1920 = 0
(x-40)(x+48)=0
x = 40 or x = -48, which of course makes no sense
there are 40 rows, and each row has 48 seats
To find the number of seats in each row, we can set up a system of equations based on the given information.
Let's assume that the number of rows is x.
According to the given information, the number of seats in each row exceeds the number of rows by 8. So, the number of seats in each row can be expressed as x + 8.
Since there are x rows and x + 8 seats in each row, the total number of seats in the rectangular auditorium can be calculated by multiplying the number of rows by the number of seats in each row:
x * (x + 8) = 1920
Now, we can solve this equation to find the value of x.
Expanding the equation, we get:
x^2 + 8x = 1920
Rearranging the equation:
x^2 + 8x - 1920 = 0
Now we have a quadratic equation. We can solve this equation by factoring, completing the square, or using the quadratic formula.
Factoring, we get:
(x - 40)(x + 48) = 0
Setting each factor equal to zero, we have:
x - 40 = 0 or x + 48 = 0
Solving for x in each case, we get:
x = 40 or x = -48
Since the number of rows cannot be negative, we discard the negative solution.
So, the number of rows in the auditorium is 40.
Finally, to find the number of seats in each row, we substitute the value of x = 40 in the equation x + 8:
40 + 8 = 48
Therefore, the number of seats in each row is 48.