The primary reason is that you have a roadblock: you absolutely need complex numbers - even for real-valued solutions - unless you go the route of using trigonometry and/or hyperbolic functions. The simplest example of the roadblock is directly connected to the ancient problem of trisecting the 60 degree angle. Write down the cubic for the cosine of 20 degrees (obtained via the triple angle formula, using the cosine of 60 degrees):
$$4c^3 - 3c = \frac{1}{2}.$$
Now, use the "Cardano" formula, and you'll quickly see the problem. In fact, there's no way to express the real root to this in terms of the four arithmetic operations and square and cube roots, without using complex numbers.
The other roadblock was negative numbers! Del Ferro solved all equations of the form $x^3 + mx = n$ before 1500 ... but only with non-negative coefficients $(m,n)$. Why? Because there wasn't any notion of negative numbers around yet!
Both complex and negative numbers came onto the scene together, not separately; because once you start talking about numbers like $-1$, you're also going to be asking questions like what $\sqrt{-1}$ is.
In the early 1500's, Tartaglia said "I have a secret"; Cardano said "Cough it up" and Tartaglia bound him to a (word-of-mouth) non-disclosure, but Cardano said: "Publish it soon or I'll end-run you" (by propping up and upgrading del Ferro's solution). Cardano's publication allowed for both negative numbers and their square roots, but left the square roots of negative numbers uninterpreted.
All of that is the account given over on the Wikipedia article Cubic Equation.
A real root to $x^3 + 3Ax + 2B = 0$ is $x = C - A/C$, where $C^3 = -B + \sqrt{A^3 + B^2}$. So, with the cosine of 20 degrees, $A = -1/4$ and $B = -1/16$, and the cosine of 20 degrees is $C + 1/(4C)$, where $C^3 = 1/16 + \sqrt{-3/256}$ ... oops. There's your problem. What's the cube root of a complex number, expressed in the form $x + i y$? How do you get the $(x,y)$ parts for $C$? Well, that involves solving a cubic equation. Guess which one?
It's the Casus Irreducibilis Impasse. According to the linked Wikipedia article, the proof that it actually is an impasse was established in 1843. So, in 300+ years' retrospect: that's your roadblock. You're forced either into complex numbers (never mind negative numbers) or - in this example - into trigonometry.
I had a similar problem, when in high school. I asked my 11th grade algebra teacher what the solution to the cubic $x^3 + x + 1 = 0$ was, and he said "You need to learn calculus for that". So, the following week, I went back to him and said, "Ok. I bought a \$5 calculus book from the bookstore, carefully read through all of the sections, did all the exercises, and learned the material, but I still can't solve the equation" (in closed form). Curiously, I never got any answer from him on that, since he was too busy fainting at the time, for some strange reason. About seven months later, the following year, I asked my third semester calculus assistant professor in college, and he finally coughed it up. So ... all that running around just to find out what the closed form solution to $x^3 + x + 1 = 0$ was. And it's not even one of those Impasse cases.
