Hardy-Weinberg for triploids
First, although it only speaks about cases of diploidy, you might want to have a look at Solving Hardy Weinberg problems.

Information given to us

The question gives the same names to the phenotypes than to the alleles which si rather confusing. Let's call the three possible phenotype $P_1$, $P_2$ and $P_3$. We know that the frequencies of these phenotypes are

$$f(P_1) = 0.614$$
$$f(P_2) = 0.306$$
$$f(P_3) = 0.08$$

Compute phenotype frequencies in terms of allele frequencies

Let, $x$, $y$ and $z$ be the frequencies of the three alleles, $A_1$, $A_2$ and $A_3$ (I chose those names to fit with the equations you wrote). The whole difficulty now is to express $f(P_1)$, $f(P_2)$ and $f(P_3)$ in terms of $x$, $y$ and $z$. You should try yourself before looking at the result. Start with $f(P_3)$, it is the easiest.

Here is the solution for $f(P_3)$


   $f(P_3) = z^3$


Here is the solution for $f(P_2)$


   $f(P_2) = y^3 + yz^2$


Here is the solution for $f(P_1)$


   $f(P_1) = x^3 + 3x^2 (y+z) + xyz + xy^2 + xz^2$


Once, you have done that, you have three equations with three unknowns and you just have to solve them. Make sure at the end that $x+y+z = f(P_1) + f(P_2) + f(P_3) = 1$.