Solve $\frac{n(n+1)}{2}= 45$ I have this equation:
$\frac{n(n+1)}{2}= 45$
I started:
$$n^2+n= 90$$
What do I do next?

$n^2+n=90$
We can write this as-
$n^2+n-90=0$
Now, it is in the quadratic equation form, .i.e., $ax^2+bx+c=0$
Factor out the resulting equation, break $n$ into two terms $10n-9n$
$$n^2+10n-9n-90=0$$
Take $n$ common from first two terms and $-9$ from last two terms-
$$n(n+10)-9(n+10)=0$$
$$(n-9)(9+10)=0$$
$n-9=0\;\implies n=9$
$n+10=0\;\implies n=-10$
