If a polygon of $n$ sides has 275 diagonals, then $n$ is equal to

The number of diagonals in a polygon of $n$ sides is given by the formula $D = \dfrac{n(n - 3)}{2}$.

Given $D = 275$, we have the equation:

$\dfrac{n(n - 3)}{2} = 275$

$n(n - 3) = 550$

$n^2 - 3n - 550 = 0$

Solving the quadratic equation using the quadratic formula $n = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

$n = \dfrac{3 \pm \sqrt{(-3)^2 - 4(1)(-550)}}{2(1)}$

$n = \dfrac{3 \pm \sqrt{9 + 2200}}{2}$

$n = \dfrac{3 \pm \sqrt{2209}}{2}$

$n = \dfrac{3 \pm 47}{2}$

Since $n$ must be positive, we take the positive root:

$n = \dfrac{3 + 47}{2} = \dfrac{50}{2} = 25$

Answer: 25