1. **State the problem:** Solve the equation $$(3x - 1)^2 - (5x - 3)^2 = (4x - 2)^2$$ for $x$. 2. **Recall the formula:** This is a difference of squares on the left side: $$a^2 - b^2 = (a - b)(a + b)$$ where $$a = 3x - 1$$ and $$b = 5x - 3$$. 3. **Apply the difference of squares:** $$ (3x - 1)^2 - (5x - 3)^2 = ((3x - 1) - (5x - 3))((3x - 1) + (5x - 3)) $$ Simplify each factor: $$ (3x - 1) - (5x - 3) = 3x - 1 - 5x + 3 = -2x + 2 $$ $$ (3x - 1) + (5x - 3) = 3x - 1 + 5x - 3 = 8x - 4 $$ So the left side becomes: $$ (-2x + 2)(8x - 4) $$ 4. **Rewrite the equation:** $$ (-2x + 2)(8x - 4) = (4x - 2)^2 $$ 5. **Expand the left side:** $$ (-2x + 2)(8x - 4) = -2x \cdot 8x + (-2x)(-4) + 2 \cdot 8x + 2 \cdot (-4) = -16x^2 + 8x + 16x - 8 = -16x^2 + 24x - 8 $$ 6. **Expand the right side:** $$ (4x - 2)^2 = (4x)^2 - 2 \cdot 4x \cdot 2 + 2^2 = 16x^2 - 16x + 4 $$ 7. **Set the equation:** $$ -16x^2 + 24x - 8 = 16x^2 - 16x + 4 $$ 8. **Bring all terms to one side:** $$ -16x^2 + 24x - 8 - 16x^2 + 16x - 4 = 0 $$ Simplify: $$ -32x^2 + 40x - 12 = 0 $$ 9. **Divide entire equation by -4 to simplify:** $$ \cancel{-4} \times (8x^2 - 10x + 3) = 0 \Rightarrow 8x^2 - 10x + 3 = 0 $$ 10. **Solve quadratic equation:** Use quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=8$$, $$b=-10$$, $$c=3$$. Calculate discriminant: $$\Delta = (-10)^2 - 4 \times 8 \times 3 = 100 - 96 = 4$$ Calculate roots: $$ x = \frac{10 \pm \sqrt{4}}{16} = \frac{10 \pm 2}{16}$$ So, $$ x_1 = \frac{10 + 2}{16} = \frac{12}{16} = \frac{3}{4}$$ $$ x_2 = \frac{10 - 2}{16} = \frac{8}{16} = \frac{1}{2}$$ **Final answer:** $$x = \frac{3}{4} \text{ or } x = \frac{1}{2}$$