1. **State the problem:** We need to find the number of extraneous solutions for the equation $$\frac{2m}{2m+3} - \frac{2m}{2m-3} = 1$$. 2. **Identify restrictions:** The denominators cannot be zero, so: $$2m+3 \neq 0 \Rightarrow m \neq -\frac{3}{2}$$ $$2m-3 \neq 0 \Rightarrow m \neq \frac{3}{2}$$ 3. **Solve the equation:** Find a common denominator and combine terms: $$\frac{2m(2m-3)}{(2m+3)(2m-3)} - \frac{2m(2m+3)}{(2m-3)(2m+3)} = 1$$ 4. Simplify the numerator: $$\frac{2m(2m-3) - 2m(2m+3)}{(2m+3)(2m-3)} = 1$$ 5. Expand the numerator: $$\frac{4m^2 - 6m - 4m^2 - 6m}{(2m+3)(2m-3)} = 1$$ 6. Simplify numerator: $$\frac{-12m}{(2m+3)(2m-3)} = 1$$ 7. Multiply both sides by the denominator: $$-12m = (2m+3)(2m-3)$$ 8. Expand the right side: $$-12m = 4m^2 - 9$$ 9. Rearrange to standard quadratic form: $$4m^2 + 12m - 9 = 0$$ 10. Solve quadratic using the quadratic formula: $$m = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} = \frac{-12 \pm \sqrt{144 + 144}}{8} = \frac{-12 \pm \sqrt{288}}{8}$$ 11. Simplify the square root: $$\sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2}$$ 12. So the solutions are: $$m = \frac{-12 \pm 12\sqrt{2}}{8} = \frac{12(-1 \pm \sqrt{2})}{8} = \frac{3(-1 \pm \sqrt{2})}{2}$$ 13. Check if solutions are extraneous by verifying they are not excluded values $m \neq \pm \frac{3}{2}$: - For $m = \frac{3(-1 + \sqrt{2})}{2}$, approximate $\sqrt{2} \approx 1.414$, so $m \approx \frac{3(-1 + 1.414)}{2} = \frac{3(0.414)}{2} = 0.621$, which is allowed. - For $m = \frac{3(-1 - \sqrt{2})}{2}$, $m \approx \frac{3(-1 - 1.414)}{2} = \frac{3(-2.414)}{2} = -3.621$, which is allowed. 14. Both solutions are valid and not extraneous. **Final answer:** The equation has **0** extraneous solutions.