In an auditorium, the number of rows is equal to the number of seats in each row. When the number of rows is doubled and the number of seats in each row is reduced by $12$, the number of seats increases by $1300$. Find the original number of rows.
Let original number of rows $= x$
Number of seats in each row $= x$
$\therefore \;$ Original total number seats $= x^2$
New number of rows $= 2x$
New number of seats $= x - 12$
$\therefore \;$ New total number seats $= 2x \left(x - 12\right) = 2x^2 - 24x$
As per sum, $\;$ New total number seats $= $ Original total number of seats $+ 1300$
i.e. $\;$ $2x^2 - 24x = x^2 + 1300$
i.e. $\;$ $x^2 - 24x - 1300 = 0$
i.e. $\;$ $x^2 - 50x + 26x - 1300 = 0$
i.e. $\;$ $x \left(x - 50\right) + 26 \left(x - 50\right) = 0$
i.e. $\;$ $\left(x + 26\right) \left(x - 50\right) = 0$
i.e. $\;$ $x + 26 = 0$ $\;$ OR $\;$ $x - 50 = 0$
i.e. $\;$ $x = -26$ $\;$ OR $\;$ $x = 50$
Since the number of rows cannot be negative,
original number of rows $= x = 50$