TITLE: Can $x^3+3x^2+1=0$ be solved using high school methods?
QUESTION [20 upvotes]: I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem:
Solve $x^3+3x^2+1=0$
Am I missing something here, or is indeed a more advanced method necessary to solve this particular cubic? The answer provided was $x\doteq-3.1$, which I was only able to confirm using CAS.

REPLY [2 votes]: I remember the Cardano's method (simplified without extension over complex numbers) described in my high school math book. We probably skipped that chapter, however that equation could be solved applying the procedure mechanically, just like a calculus exercise.
However, given the fact that the solution is given as a simple number and not as a radical, I imagine that an approximation like the one listed above was required.