Solve the equation. Express numbers in exact simplest form. \[ q^{3}+9 q=6 q^{2} \] The solution set is \( \{\square\} \). \( \square \square, \square, \ldots \frac{\square}{\square} \)

We are given the equation \[ q^{3} + 9q = 6q^{2}. \] First, we bring all terms to one side: \[ q^{3} + 9q - 6q^{2} = 0. \] Rearrange the terms: \[ q^{3} - 6q^{2} + 9q = 0. \] Notice that each term has a common factor \(q\). Factor it out: \[ q\left(q^{2} - 6q + 9\right) = 0. \] Now, the quadratic inside the parentheses can be factored further. Recognize that \[ q^{2} - 6q + 9 = \left(q - 3\right)^{2}, \] because expanding \(\left(q - 3\right)^{2}\) gives \(q^{2} - 6q+9\). Thus, the complete factorization of the original equation is: \[ q\left(q-3\right)^{2} = 0. \] We now set each factor equal to zero: 1. \(q = 0\). 2. \(\left(q-3\right)^{2} = 0 \implies q - 3 = 0 \implies q = 3\). Therefore, the solutions are \(q=0\) and \(q=3\). The solution set in exact simplest form is \[ \{0,\, 3\}. \]