1. **State the problem:** Solve the equation $$\frac{(Cx - 1)^2}{3} - \frac{x - 2}{2} = 2$$ for $x$. 2. **Rewrite the equation:** Multiply both sides by the least common denominator (LCD) of 6 to clear fractions: $$6 \times \left(\frac{(Cx - 1)^2}{3} - \frac{x - 2}{2}\right) = 6 \times 2$$ 3. **Simplify each term:** $$2 (Cx - 1)^2 - 3 (x - 2) = 12$$ 4. **Expand the squared term:** $$(Cx - 1)^2 = (Cx)^2 - 2 \cdot Cx \cdot 1 + 1^2 = C^2 x^2 - 2 C x + 1$$ 5. **Substitute back:** $$2 (C^2 x^2 - 2 C x + 1) - 3 (x - 2) = 12$$ 6. **Distribute:** $$2 C^2 x^2 - 4 C x + 2 - 3 x + 6 = 12$$ 7. **Combine like terms:** $$2 C^2 x^2 - (4 C + 3) x + 8 = 12$$ 8. **Bring all terms to one side:** $$2 C^2 x^2 - (4 C + 3) x + 8 - 12 = 0$$ $$2 C^2 x^2 - (4 C + 3) x - 4 = 0$$ 9. **Use the quadratic formula:** For equation $$a x^2 + b x + c = 0$$, solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$$ Here, $$a = 2 C^2$$, $$b = -(4 C + 3)$$, $$c = -4$$. 10. **Calculate discriminant:** $$\Delta = b^2 - 4 a c = (-(4 C + 3))^2 - 4 \times 2 C^2 \times (-4) = (4 C + 3)^2 + 32 C^2$$ 11. **Write the solutions:** $$x = \frac{4 C + 3 \pm \sqrt{(4 C + 3)^2 + 32 C^2}}{4 C^2}$$ **Final answer:** $$\boxed{x = \frac{4 C + 3 \pm \sqrt{(4 C + 3)^2 + 32 C^2}}{4 C^2}}$$ This gives the two possible values of $x$ depending on the sign chosen in the quadratic formula.