1. **State the problem:** Simplify and solve the equation $$(3x-1)^2-(x-1) = (9x-1)+(3x-1).$$ 2. **Expand the squares and simplify each side:** $$(3x-1)^2 = (3x)^2 - 2 \cdot 3x \cdot 1 + 1^2 = 9x^2 - 6x + 1.$$ So the left side becomes: $$9x^2 - 6x + 1 - (x - 1) = 9x^2 - 6x + 1 - x + 1 = 9x^2 - 7x + 2.$$ The right side is: $$(9x - 1) + (3x - 1) = 9x - 1 + 3x - 1 = 12x - 2.$$ 3. **Set the equation:** $$9x^2 - 7x + 2 = 12x - 2.$$ Bring all terms to one side: $$9x^2 - 7x + 2 - 12x + 2 = 0,$$ which simplifies to $$9x^2 - 19x + 4 = 0.$$ 4. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=9$, $b=-19$, and $c=4$. Calculate the discriminant: $$\Delta = (-19)^2 - 4 \cdot 9 \cdot 4 = 361 - 144 = 217.$$ 5. **Write the solutions:** $$x = \frac{19 \pm \sqrt{217}}{18}.$$ **Final answer:** $$x = \frac{19 + \sqrt{217}}{18} \quad \text{or} \quad x = \frac{19 - \sqrt{217}}{18}.$$