Question

Find the number of sides of a polygon which has 14 diagonals.

Answer 1

Number of diagonals of a polygon of $n$ sides $=\frac{n(n-1)}{2}-n$
$\Rightarrow 14=\frac{n(n-1)}{2}-n$
$\Rightarrow 28=n^2-n-2n$
$\Rightarrow n^2-3n-28=0$
$\Rightarrow n^2-7n+4n-28=0$
$\Rightarrow n(n-7)+4(n-7)=0$
$\Rightarrow (n-7)(n+4)=0$
$\Rightarrow n=7$