Question: How would I solve $3x^2 \times \left(\frac{\sqrt{3}}{2}\right) = 18 \sqrt{3}$?
1. **State the problem:** Solve the equation $$3x^2 \times \left(\frac{\sqrt{3}}{2}\right) = 18 \sqrt{3}$$ for $x$.
2. **Write the formula and rules:** We want to isolate $x$. The equation involves multiplication and a square term $x^2$. We will first simplify and then solve for $x^2$.
3. **Simplify the left side:**
$$3x^2 \times \frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{2} x^2$$
4. **Rewrite the equation:**
$$\frac{3 \sqrt{3}}{2} x^2 = 18 \sqrt{3}$$
5. **Divide both sides by $\frac{3 \sqrt{3}}{2}$ to isolate $x^2$:**
$$x^2 = \frac{18 \sqrt{3}}{\frac{3 \sqrt{3}}{2}}$$
6. **Simplify the division by multiplying by the reciprocal:**
$$x^2 = 18 \sqrt{3} \times \frac{2}{3 \sqrt{3}}$$
7. **Cancel common factors:**
$$x^2 = 18 \cancel{\sqrt{3}} \times \frac{2}{3 \cancel{\sqrt{3}}} = 18 \times \frac{2}{3}$$
8. **Calculate the right side:**
$$x^2 = \frac{18 \times 2}{3} = \frac{36}{3} = 12$$
9. **Take the square root of both sides:**
$$x = \pm \sqrt{12} = \pm 2 \sqrt{3}$$
**Final answer:**
$$x = \pm 2 \sqrt{3}$$