1. **State the problem:** Solve the equation $x^2 + \frac{3}{2}x = 0$. 2. **Formula and rules:** To solve quadratic equations, we can factorize or use the quadratic formula. Here, factoring is straightforward. 3. **Rewrite the equation:** $$x^2 + \frac{3}{2}x = 0$$ 4. **Factor out the common term $x$:** $$x\left(x + \frac{3}{2}\right) = 0$$ 5. **Apply the zero product property:** Set each factor equal to zero: $$x = 0$$ $$x + \frac{3}{2} = 0$$ 6. **Solve each equation:** - For $x = 0$, solution is $x=0$. - For $x + \frac{3}{2} = 0$, subtract $\frac{3}{2}$ from both sides: $$x + \cancel{\frac{3}{2}} = 0 - \cancel{\frac{3}{2}}$$ $$x = -\frac{3}{2}$$ 7. **Final solutions:** $$x = 0 \quad \text{or} \quad x = -\frac{3}{2}$$