1. **State the problem:** Solve the equation $$(3x-1)^2-(x-1) = (9x-3)+(3x-1).$$ 2. **Expand and simplify:** Expand the square term: $$ (3x-1)^2 = (3x)^2 - 2 \cdot 3x \cdot 1 + 1^2 = 9x^2 - 6x + 1. $$ Rewrite the equation: $$ 9x^2 - 6x + 1 - (x - 1) = 9x - 3 + 3x - 1. $$ Simplify inside the parentheses: $$ 9x^2 - 6x + 1 - x + 1 = 9x - 3 + 3x - 1. $$ Combine like terms on the left: $$ 9x^2 - 7x + 2 = 12x - 4. $$ 3. **Bring all terms to one side:** $$ 9x^2 - 7x + 2 - 12x + 4 = 0 $$ Simplify: $$ 9x^2 - 19x + 6 = 0. $$ 4. **Solve the quadratic equation:** Use the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=9$, $b=-19$, $c=6$. Calculate the discriminant: $$ \Delta = (-19)^2 - 4 \cdot 9 \cdot 6 = 361 - 216 = 145. $$ 5. **Find the roots:** $$ x = \frac{19 \pm \sqrt{145}}{18}. $$ 6. **Final answer:** $$ x = \frac{19 + \sqrt{145}}{18} \quad \text{or} \quad x = \frac{19 - \sqrt{145}}{18}. $$