1. **State the problem:** Solve the equation $$(x - 3)^2(x^2 - 3) = 0$$ for $x$. 2. **Recall the zero product property:** If a product of factors equals zero, then at least one of the factors must be zero. So, set each factor equal to zero: $$ (x - 3)^2 = 0 \quad \text{or} \quad x^2 - 3 = 0 $$ 3. **Solve the first factor:** $$ (x - 3)^2 = 0 \implies x - 3 = 0 \implies x = 3 $$ 4. **Solve the second factor:** $$ x^2 - 3 = 0 \implies x^2 = 3 \implies x = \pm \sqrt{3} $$ 5. **Final solution:** The roots of the equation are $$ x = 3, \quad x = \sqrt{3}, \quad x = -\sqrt{3} $$ These are the points where the graph intersects the x-axis. Note that $x=3$ is a root with multiplicity 2, meaning the graph touches the x-axis there but does not cross it.