1. **Problem a:** Solve the equation $$\frac{6}{x - 1} = \frac{x}{x^2 + 2x + 1}$$ 2. **Step 1:** Recognize that $$x^2 + 2x + 1 = (x + 1)^2$$. 3. **Step 2:** Cross-multiply to eliminate denominators: $$6(x + 1)^2 = x(x - 1)$$ 4. **Step 3:** Expand both sides: $$6(x^2 + 2x + 1) = x^2 - x$$ $$6x^2 + 12x + 6 = x^2 - x$$ 5. **Step 4:** Bring all terms to one side: $$6x^2 + 12x + 6 - x^2 + x = 0$$ $$5x^2 + 13x + 6 = 0$$ 6. **Step 5:** Solve the quadratic equation using the quadratic formula: $$x = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 5 \cdot 6}}{2 \cdot 5} = \frac{-13 \pm \sqrt{169 - 120}}{10} = \frac{-13 \pm \sqrt{49}}{10}$$ 7. **Step 6:** Calculate the roots: $$x = \frac{-13 + 7}{10} = \frac{-6}{10} = -0.6$$ $$x = \frac{-13 - 7}{10} = \frac{-20}{10} = -2$$ 8. **Step 7:** Check for restrictions: denominators cannot be zero. - For $$x - 1 \neq 0$$, so $$x \neq 1$$. - For $$(x + 1)^2 \neq 0$$, so $$x \neq -1$$. Both solutions $$x = -0.6$$ and $$x = -2$$ are valid. --- 9. **Problem b:** Solve the equation $$\frac{2}{x^2 - 1} + \frac{1}{x + 1} = 1$$ 10. **Step 1:** Factor denominator: $$x^2 - 1 = (x - 1)(x + 1)$$ 11. **Step 2:** Find common denominator $$ (x - 1)(x + 1) $$ and rewrite terms: $$\frac{2}{(x - 1)(x + 1)} + \frac{1}{x + 1} = 1$$ 12. **Step 3:** Express $$\frac{1}{x + 1}$$ with common denominator: $$\frac{1}{x + 1} = \frac{x - 1}{(x - 1)(x + 1)}$$ 13. **Step 4:** Combine fractions: $$\frac{2 + (x - 1)}{(x - 1)(x + 1)} = 1$$ $$\frac{x + 1}{(x - 1)(x + 1)} = 1$$ 14. **Step 5:** Simplify numerator and denominator: $$\frac{x + 1}{(x - 1)(x + 1)} = \frac{1}{x - 1} = 1$$ 15. **Step 6:** Solve for $$x$$: $$\frac{1}{x - 1} = 1 \implies x - 1 = 1 \implies x = 2$$ 16. **Step 7:** Check restrictions: $$x^2 - 1 \neq 0 \implies x \neq \pm 1$$ Solution $$x = 2$$ is valid. --- 17. **Problem 13:** Solve $$x\sqrt{7} = x\sqrt{2} + 4\sqrt{2}$$ 18. **Step 1:** Rearrange terms: $$x\sqrt{7} - x\sqrt{2} = 4\sqrt{2}$$ 19. **Step 2:** Factor $$x$$: $$x(\sqrt{7} - \sqrt{2}) = 4\sqrt{2}$$ 20. **Step 3:** Solve for $$x$$: $$x = \frac{4\sqrt{2}}{\sqrt{7} - \sqrt{2}}$$ 21. **Step 4:** Rationalize denominator: $$x = \frac{4\sqrt{2}(\sqrt{7} + \sqrt{2})}{(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})} = \frac{4\sqrt{2}(\sqrt{7} + \sqrt{2})}{7 - 2} = \frac{4\sqrt{2}(\sqrt{7} + \sqrt{2})}{5}$$ 22. **Step 5:** Simplify numerator: $$4\sqrt{2} \cdot \sqrt{7} = 4\sqrt{14}$$ $$4\sqrt{2} \cdot \sqrt{2} = 4 \times 2 = 8$$ 23. **Step 6:** Write final answer: $$x = \frac{8 + 4\sqrt{14}}{5}$$ --- **Final answers:** - a) $$x = -0.6, -2$$ - b) $$x = 2$$ - 13) $$x = \frac{8 + 4\sqrt{14}}{5}$$