1. **State the problem:** Solve the equation $$\frac{x}{x+2} - \frac{7}{x-2} = 0$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator and combine the fractions. Then solve the resulting equation. 3. **Find the common denominator:** The denominators are $x+2$ and $x-2$. The common denominator is $$(x+2)(x-2)$$. 4. **Rewrite each fraction with the common denominator:** $$\frac{x}{x+2} = \frac{x(x-2)}{(x+2)(x-2)}$$ $$\frac{7}{x-2} = \frac{7(x+2)}{(x+2)(x-2)}$$ 5. **Set up the equation:** $$\frac{x(x-2)}{(x+2)(x-2)} - \frac{7(x+2)}{(x+2)(x-2)} = 0$$ 6. **Combine the fractions:** $$\frac{x(x-2) - 7(x+2)}{(x+2)(x-2)} = 0$$ 7. **For a fraction to be zero, the numerator must be zero:** $$x(x-2) - 7(x+2) = 0$$ 8. **Expand the numerator:** $$x^2 - 2x - 7x - 14 = 0$$ $$x^2 - 9x - 14 = 0$$ 9. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-9$, $c=-14$. 10. **Calculate the discriminant:** $$\Delta = (-9)^2 - 4(1)(-14) = 81 + 56 = 137$$ 11. **Find the roots:** $$x = \frac{9 \pm \sqrt{137}}{2}$$ 12. **Check for restrictions:** The denominators $x+2$ and $x-2$ cannot be zero, so $x \neq -2$ and $x \neq 2$. 13. **Final answer:** $$x = \frac{9 + \sqrt{137}}{2} \quad \text{or} \quad x = \frac{9 - \sqrt{137}}{2}$$ Both values are valid since neither equals $-2$ or $2$.