1. **State the problem:** Solve the equation $$24x^2 - 2x^4 = 0$$ for $x$. 2. **Rewrite the equation:** Factor out the common term. $$24x^2 - 2x^4 = 2x^2(12 - x^2) = 0$$ 3. **Apply the zero product property:** For a product to be zero, at least one factor must be zero. Set each factor equal to zero: $$2x^2 = 0 \quad \Rightarrow \quad x^2 = 0$$ $$12 - x^2 = 0 \quad \Rightarrow \quad x^2 = 12$$ 4. **Solve each equation:** From $x^2 = 0$, we get: $$x = 0$$ From $x^2 = 12$, take the square root of both sides: $$x = \pm \sqrt{12}$$ Simplify the square root: $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ 5. **Final solutions:** $$x = 0, \quad x = 2\sqrt{3}, \quad x = -2\sqrt{3}$$