The number of seats in a row has been increased at a rectangular-shape movie theater after renovation so that it is now five more than the total number of rows. How many rows are in in the theater, if its total seating capacity is 126 people now?
To find the number of rows in the theater, let's assume the number of rows is "x".
According to the given information, the number of seats in a row has been increased by 5 from the total number of rows. So, the number of seats in a row is now (x + 5).
Since a rectangular-shaped theater has the same number of seats in each row, we can calculate the total number of seats in the theater as the product of the number of rows and the number of seats in a row.
Total seating capacity = Number of rows × Number of seats in a row
126 = x × (x + 5)
Now, let's solve this quadratic equation for x:
x² + 5x = 126
Rearranging the equation:
x² + 5x - 126 = 0
Now we need to factorize or use the quadratic formula to solve this equation. Let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In this equation, a = 1, b = 5, and c = -126. Substituting these values into the formula:
x = (-5 ± √(5² - 4(1)(-126))) / 2(1)
Simplifying further:
x = (-5 ± √(25 + 504)) / 2
x = (-5 ± √529) / 2
We can now calculate the two possible values for x:
x₁ = (-5 + √529) / 2
x₂ = (-5 - √529) / 2
x₁ = (-5 + 23) / 2 = 18 / 2 = 9
x₂ = (-5 - 23) / 2 = -28 / 2 = -14
Since the number of rows cannot be negative, we discard the negative value. Therefore, there are 9 rows in the theater.
so, what do you know? If there are n rows, then
n(n+5) = 126
hint: 9*14 = 126