## Elementary Geometry for College Students (6th Edition)

The number of sides in the polygon is $~6$
Let's consider a trinomial in this form: $~x^2+bx+c$ To factor this trinomial, we need to find two numbers $r$ and $s$ such that $r+s = b$ and $r\times s = c$ Then the next step is to rewrite the trinomial as follows: $~x^2+bx+c = (x+r)~(x+s)$ We can rewrite the given equation with $D=9$: $\frac{n~(n-3)}{2} = D$ $\frac{n~(n-3)}{2} = 9$ $n~(n-3) = 18$ $n^2-3n = 18$ $n^2-3n - 18 = 0$ To factor the left side of this equation, we need to find two numbers $r$ and $s$ such that $r+s = -3~$ and $~r\times s = -18$. We can see that $(-6)+(3) = -3~$ and $(-6)\times (3) = -18$ We can solve the equation as follows: $\frac{n~(n-3)}{2} = D$ $\frac{n~(n-3)}{2} = 9$ $n~(n-3) = 18$ $n^2-3n = 18$ $n^2-3n - 18 = 0$ $(n-6)~(n+3) = 0$ $n-6=0~~$or $~~n+3=0$ $n = 6~~$ or $~~n=-3$ Since the number $n$ must be a positive number, $n=6$ The number of sides in the polygon is $~6$