1. Solve the equation $25x^4 - 104x^2 + 16 = 0$. 2. Start by using substitution $u = x^2$ to simplify the quartic to a quadratic in $u$: $$25u^2 - 104u + 16 = 0$$ 3. Calculate the discriminant: $$\Delta = (-104)^2 - 4 \cdot 25 \cdot 16 = 10816 - 1600 = 9216$$ 4. Take the square root: $$\sqrt{9216} = 96$$ 5. Use the quadratic formula to find $u$: $$u = \frac{104 \pm 96}{2 \cdot 25} = \frac{104 \pm 96}{50}$$ 6. Calculate both roots: $$u_1 = \frac{104 + 96}{50} = \frac{200}{50} = 4$$ $$u_2 = \frac{104 - 96}{50} = \frac{8}{50} = \frac{4}{25}$$ 7. Recall $u = x^2$, so solve: $$x^2 = 4 \Rightarrow x = \pm 2$$ $$x^2 = \frac{4}{25} \Rightarrow x = \pm \frac{2}{5}$$ --- 8. Solve equation 2: $\frac{y - c}{y + c} + \frac{y + c}{y - c} = \frac{34}{15}$ 9. Find common denominator and simplify left side by letting $A = \frac{y - c}{y + c}$ and $B = \frac{y + c}{y - c}$: $$A + B = \frac{(y - c)^2 + (y + c)^2}{(y + c)(y - c)}$$ 10. Expand numerator: $$(y - c)^2 = y^2 - 2yc + c^2$$ $$(y + c)^2 = y^2 + 2yc + c^2$$ Sum: $$2y^2 + 2c^2$$ 11. Denominator: $$(y + c)(y - c) = y^2 - c^2$$ 12. So the equation becomes: $$\frac{2y^2 + 2c^2}{y^2 - c^2} = \frac{34}{15}$$ 13. Cross multiply: $$15(2y^2 + 2c^2) = 34(y^2 - c^2)$$ 14. Simplify: $$30y^2 + 30c^2 = 34y^2 - 34c^2$$ 15. Bring terms to one side: $$30y^2 - 34y^2 + 30c^2 + 34c^2 = 0$$ $$-4y^2 + 64c^2 = 0$$ 16. Solve for $y^2$: $$4y^2 = 64c^2 \Rightarrow y^2 = 16c^2 \Rightarrow y = \pm 4c$$ --- 17. Solve equation 3: $\frac{x - 3}{x + 3} + \frac{x + 3}{x - 3} = \frac{2 \times 4}{15} = \frac{8}{15}$ 18. Similarly, let $A = \frac{x - 3}{x + 3}$ and $B = \frac{x + 3}{x - 3}$. 19. Calculate numerator: $$(x - 3)^2 + (x + 3)^2 = (x^2 - 6x + 9) + (x^2 + 6x + 9) = 2x^2 + 18$$ 20. Denominator: $$(x + 3)(x - 3) = x^2 - 9$$ 21. The equation is: $$\frac{2x^2 + 18}{x^2 - 9} = \frac{8}{15}$$ 22. Cross multiply: $$15(2x^2 + 18) = 8(x^2 - 9)$$ 23. Simplify: $$30x^2 + 270 = 8x^2 - 72$$ 24. Bring all to left: $$30x^2 - 8x^2 + 270 + 72 = 0$$ $$22x^2 + 342 = 0$$ 25. Solve for $x^2$: $$22x^2 = -342 \Rightarrow x^2 = -\frac{342}{22} = -\frac{171}{11}$$ 26. Since $x^2$ is negative, there is no real solution for $x$. --- 27. Solve equation 4: $\sqrt{\frac{t + 16}{t}} + \sqrt{\frac{t}{t + 16}} = \frac{2 \times 1}{12} = \frac{1}{6}$ 28. Let $a = \sqrt{\frac{t + 16}{t}}$ and $b = \sqrt{\frac{t}{t + 16}}$. 29. Notice $ab = \sqrt{1} = 1$. 30. The equation becomes: $$a + b = \frac{1}{6}$$ 31. Square both sides: $$(a + b)^2 = \left( \frac{1}{6} \right)^2$$ $$a^2 + 2ab + b^2 = \frac{1}{36}$$ 32. Since $ab = 1$, substitute: $$a^2 + 2(1) + b^2 = \frac{1}{36}$$ 33. Recall $a^2 = \frac{t+16}{t}$ and $b^2 = \frac{t}{t+16}$: $$\frac{t + 16}{t} + \frac{t}{t + 16} + 2 = \frac{1}{36}$$ 34. Combine left side terms as one fraction: $$\frac{(t + 16)^2 + t^2}{t(t + 16)} + 2 = \frac{1}{36}$$ 35. Expand numerator: $$(t + 16)^2 + t^2 = t^2 + 32t + 256 + t^2 = 2t^2 + 32t + 256$$ 36. Hence: $$\frac{2t^2 + 32t + 256}{t(t + 16)} + 2 = \frac{1}{36}$$ 37. Rewrite 2 as $\frac{2t(t + 16)}{t(t + 16)}$: $$\frac{2t^2 + 32t + 256 + 2t^2 + 32t}{t(t + 16)} = \frac{1}{36}$$ $$\frac{4t^2 + 64t + 256}{t(t + 16)} = \frac{1}{36}$$ 38. Cross multiply: $$36(4t^2 + 64t + 256) = t(t + 16)$$ 39. Expand left: $$144t^2 + 2304t + 9216 = t^2 + 16t$$ 40. Bring all terms to one side: $$144t^2 - t^2 + 2304t - 16t + 9216 = 0$$ $$143t^2 + 2288t + 9216 = 0$$ 41. Use quadratic formula: $$t = \frac{-2288 \pm \sqrt{2288^2 - 4 \cdot 143 \cdot 9216}}{2 \cdot 143}$$ 42. Calculate discriminant: $$2288^2 = 5234944$$ $$4 \cdot 143 \cdot 9216 = 5269248$$ 43. Subtract: $$5234944 - 5269248 = -34304$$ 44. Discriminant is negative, so no real solutions for $t$. --- **Final answers:** 1. $x = \pm 2, \pm \frac{2}{5}$ 2. $y = \pm 4c$ 3. No real solution for $x$ 4. No real solution for $t$