Can any real/complex number be expressed in the form of $l(l+1)$?

Question

I am learning about differential equations and a book claimed that writing an unknown constant, which can possibly be a complex number, as $l(l+1)$ has no loss of generality if we require that $l$ can be complex.

How does one show that any real or complex number can be written in the form $l(l+1)$, where $l$ can be complex?

Answer 1

We need to solve

$$l(l+1)=z \iff l^2+l-z=0 $$

and therefore

$$l=\frac{-1\pm\sqrt{1+4z}}{2}$$

Answer 2

As mentioned, one can use the quadratic formula for this. More generally, any polynomial $\Bbb C \to \Bbb C$ is surjective by the Fundamental Theorem of Algebra.