# 49. Quadratic Equations Problems Part 6¶

1. If $$a, b, c$$ are rational, show that the roots of equation $$abc^2x^2 + 3a^2cx + b^2cx - 6a^2 - ab + 2b^2 = 0$$ are rational.

2. If the roots of the equation $$ax^2 + bx + c = 0$$ be in the ration $$m:n$$ prove that $$\sqrt{\frac{m}{n}} + \sqrt{\frac{n}{m}} + \frac{b}{\sqrt{ac}} = 0$$

3. If one root of the equation $$x^2 + x.f(a) + a = 0$$ is equal to the third power of the other, determine the function $$f(x)$$.

4. If $$\alpha, \beta$$ are the roots of the equation $$x^2 - px + q = 0,$$ then find the quadratic equation the roots of which are $$(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$$ and $$\alpha^3\beta^2 + \alpha^2\beta^3$$

5. If $$\alpha, \beta$$ are the roots of the equation $$x^2 - bx + c = 0,$$ then find the quadratic equation whose roots are $$(\alpha^2 + \beta^2)(\alpha^3 + \beta^3)$$ and $$\alpha^5\beta^3 + \alpha^3\beta^5 - 2\alpha^4\beta^4$$.

6. If the sum of the roots of the quadratic equation $$ax^2 + bx + c = 0$$ is equal to the sum of the squares of their reciprocals, then show that $$\frac{b^2}{ac} + \frac{bc}{a^2} = 2.$$

7. The time of oscillation of a rigid body about a horizontal axis at a distance $$h$$ from the C. G. is given by $$T = 2\pi \sqrt{\frac{h^2 + k^2}{gh}},$$ where $$k$$ is a constant. Show that there are two values for $$h$$ for a given value of $$T$$. If $$h_1$$ and $$h_2$$ are two values of $$h,$$ show that $$h_1 + h_2 = \frac{gt^2}{4\pi^2}$$ and $$h_1h_2 = k^2.$$

8. If $$\alpha_1,\alpha_2$$ be the roots of the equation $$x^2 + px + q = 0$$ and $$\beta_1,\beta_2$$ be roots of $$x^2 + rx + s = 0$$ and the system of equations $$\alpha_1y + \alpha_2z = 0$$ and $$\beta_1y + \beta_2z = 0$$ has non trivial solutions then show that $$\frac{p^2}{r^2} = \frac{q}{s}$$

9. If $$a, b, c$$ are in H. P. and $$\alpha, \beta$$ be the roots of $$ax^2 + bx + c = 0,$$ show that $$-(1 + \alpha\beta)$$ is the H. M. of $$\alpha$$ and $$\beta.$$

10. If $$\alpha, \beta$$ are the roots of the equation $$x + 1 = \lambda x(1 - \lambda x)$$ and if $$\lambda_1, lambda_2$$ are the two values of $$\lambda$$ determined from the equation $$\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = r - 2,$$ show that $$\frac{\lambda_1^2}{\lambda_2^2} + \frac{\lambda_2^2}{\lambda_1^2} + 2 = 4\left(\frac{r + 1}{r - 1}\right)^2$$

11. If the roots of equation $$ax^2 + bx + c = 0$$ are reciprocals of those of $$lx^2 + mx + n = 0,$$ then prove that $$a:b:c = n:m:l,$$ where $$a, b, c, l, m, n$$ are all non zero.

12. If $$x_1, x_2$$ be the roots of the equation $$x^2 - 3x + A = 0$$ and $$x_3, x_4$$ be those of the equation $$x^2 - 12x + B = 0$$ and $$x_1, x_2, x_3, x_4$$ be an increasing G. P. find $$A$$ and $$B$$.

13. Let $$p$$ and $$q$$ be roots of the equation $$x^2 - 2x + A = 0$$ and let $$r$$ and $$s$$ be the roots of the equation $$x^2 - 18x + B = 0.$$ If $$p < q < r < s$$ are in A. P. find the values of $$A$$ and $$B.$$

14. Let $$\alpha, \beta$$ be the roots of the equation $$x^2 + ax - \frac{1}{2a^2} = 0, a$$ being a real parameter, prove that $$\alpha^4 + \beta^4 \ge 2 + \sqrt{2}.$$

15. If $$\alpha, \beta$$ be the roots of the equation $$x^2 - px + q = 0$$ and $$\alpha > 0, \beta > 0,$$ then find the value of $$\alpha^{\frac{1}{4}} + \beta^{\frac{1}{4}}.$$

16. If the difference between roots of the equation $$ax^2 - bx + c = 0$$ is same as the difference between the roots of equation $$bx^2 - cx + a = 0,$$ then show that $$b^4 - a^2c^2 = 4ab(bc - a^2).$$

17. If $$f(x) = 0$$ is a cubic equation with real roots $$\alpha, \beta, \gamma$$ in order of magnitudes, show that one root of equation $$f'(x) = 0$$ lies between $$\frac{1}{2}(\alpha + \beta)$$ and $$\frac{1}{2}(2\alpha + \beta)$$ and the other root lies between $$\frac{1}{2}(\beta + \gamma)$$ and $$\frac{1}{3}(2\beta + \gamma).$$

18. Let $$D_1$$ be the discriminant and $$\alpha, \beta$$ be the roots of the equation $$ax^2 + bx + c = 0$$ and $$D_2$$ be the discriminant and $$\gamma, \delta$$ be the roots of the equation $$px^2 + qx + r = 0.$$ If $$\alpha, \beta, \gamma, \delta$$ are in A. P. then prove that $$D_1:D_2 = a^2:p^2.$$

19. If $$\alpha, \beta$$ be the roots of the equation $$ax^2 + bx + c = 0$$ and $$\alpha + h, \beta + h$$ be those of equation $$px^2 + qx + r = 0,$$ then show that $$\frac{b^2 - 4ac}{a^2} = \frac{q^2 - 4pr}{p^2}.$$

20. If $$\alpha, \beta$$ be the roots of the equation $$ax^2 + bx + c = 0$$ and $$\alpha + h, \beta + h$$ be those of equation $$px^2 + qx + r = 0,$$ then show that $$2h = \frac{b}{a} - \frac{q}{p}$$

21. If $$\alpha, \beta$$ be the real and distinct roots of equation $$ax^2 + bx + c = 0$$ and $$\alpha^4, \beta^4$$ be those of equation $$lx^2 + mx + n = 0,$$ prove that the roots of equation $$a^2lx^2 - 4aclx + 2c^2l + a^2m = 0$$ are real and opposite in sign.

22. If $$\alpha, \beta$$ be the roots of equation $$ax^2 + bx + c = 0$$ and $$\gamma, \delta$$ those of equation $$lx^2 + mx + n = 0,$$ then find the equation whose roots are $$\alpha\gamma + \beta\delta$$ and $$\alpha\delta + \beta\gamma.$$

23. If $$p, q$$ be the roots of the equation $$x^2 + bx + c = 0,$$ prove that $$b$$ and $$c$$ are the roots of the equation $$x^2 + (p + q - pq)x - pq(p + q) = 0.$$

24. If $$3p^2 = 5p + 2$$ and $$3q^2 = 5q + 2$$ where $$p \ne q,$$ obtain the equation whose roots are $$3p - 2q$$ and $$3q - 2p.$$

25. If $$\alpha \pm \sqrt{\beta}$$ be the roots of the equation $$x^2 + px + q = 0,$$ prove that $$\frac{1}{\alpha} \pm \frac{1}{\sqrt{\beta}}$$ will be the roots of the equation $$(p^2 - 4q)(p^2x^2 + 4px) = 16q$$

26. If $$\alpha, \beta$$ be the roots of the equation $$x^2 - px + q = 0,$$ form the equation whose roots are $$\alpha^2\left(\frac{\alpha^2}{\beta} - \beta\right)$$ and $$\beta^2\left(\frac{\beta^2}{\alpha} - \alpha\right)$$

27. Let $$a, b, c, d$$ be real numbers in G. P. If $$u, v, w$$ satisfy the system of equations $$u + 2v + 3w = 6, 4u + 5v + 6w = 12, 6u + 9v = 4$$ then show that the roots of the equation $$\left(\frac{1}{u} + \frac{1}{v} + \frac{1}{w}\right)x^2 + [(b - c)^2 + (c - a)^2 + (d - b)^2]x + u + v + w = 0$$ and $$20x^2 + 10(a - d)^2x - 9 = 0$$ are reciprocals of each other.

28. If $$\alpha_1, \alpha_2, ..., \alpha_n$$ be the roots of the equation $$(\beta_1 - x)(\beta_2 - x) ... (\beta_n - x) + A = 0,$$ find the equation whose roots are $$\beta_1, \beta_2, ..., \beta_n.$$

29. If $$\alpha_1, \alpha_2, ..., \alpha_n$$ be the roots of the equation $$x^n + nax - b = 0,$$ show that $$(\alpha_1 - \alpha_2)(\alpha_1 - \alpha_3) ... (\alpha_1 - \alpha_n) = n(\alpha^{n - 1} + a)$$

30. If $$\alpha, \beta, \gamma, \delta$$ be the real roots of equation $$x^4 + qx^2 + rx + t = 0,$$ find the quadratic equation whose roots are $$(1 + \alpha^2)(1 + \beta^2)(1 + \gamma^2)(1 + \delta^2)$$ and $$1.$$

31. If $$\alpha, \beta, \gamma$$ be the roots of the equation $$x^3 + px + q = 0,$$ find the cubic equation whose roots are $$\frac{\alpha + 1}{\alpha}, \frac{\beta + 1}{\beta}, \frac{\gamma + 1}{\gamma}.$$

32. Show that one of the roots of the equation $$ax^2 + bx + c = 0$$ may be reciprocal of one of the roots of $$a_1x^2 + b_1x + c_1 = 0$$ if $$(aa_1 - cc_1)^2 = (hc_1 - ab_1)(b_1c - a_1b)$$

33. If every pair of the equations $$x^2 + px + qr = 0, x^2 + qx + pr = 0$$ and $$x^2 + rx + pq = 0$$ have a common root, find the sum of the three common roots.

34. If equation $$a^2(b^2 - c^2)x^2 + b^2(c^2 - a^2)x + c^2(a^2 - b^2) = 0$$ has equal roots and equations $$4x^2\sin^2\theta - 4\sin\theta + 1 = 0$$ and they have a common root then, find the value of $$\theta.$$

35. If $$a \ne 0,$$ find the value of $$a$$ for which one of the roots of equation $$x^2 - x + 3a = 0$$ is double the roots of the quadratic equation $$x^2 - x + a = 0.$$

36. If by eliminating $$x$$ between the equations $$x^2 + ax + b = 0$$ and $$xy + l(x + y) + m = 0,$$ a quadratic equation in terms of $$y$$ is formed whose roots are same as those of original quadratic equation in $$x,$$ then prove that either $$a = 2l$$ or $$b = m$$ or $$b + m = al.$$

37. The roots of equation $$10x^3 - cx^2 - 54x - 27 = 0$$ are in H. P., then find $$c.$$

38. If $$a, b, c$$ are the roots of the equation $$x^3 + px^2 + qx + r = 0$$ such that $$c^2 = -ab,$$ show that $$(2q - p^2)^3.r = (pq - 4r)^3$$

39. Let $$\alpha + i\beta, \alpha, \beta \in R$$ be roots of the equation $$x^3 + qx + r = 0, q, r \in R.$$ Find a real cubic equation independent of $$\alpha$$ and $$\beta,$$ whose one root is $$2\alpha.$$

40. If $$\alpha, \beta, \gamma$$ be the roots of the equation $$2x^3 + x^2 - 7 = 0,$$ show that $$\sum \left(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right) = -3$$

41. The equations $$x^3 + px^2 + qx + r = 0$$ and $$x^3 + p'x^2 + q'x + r' = 0$$ have two common roots, find the quadratic equations whose roots are these common roots.

42. Find the condition that the roots of equation $$ax^3 + 3bx^2 + 3cx + d = 0$$ may be in G. P.

43. Find the condition that the equation $$x^3 - px^2 + qx - r = 0$$ have its roots in H. P.

44. If $$f(x) = x^3 + bx^2 + cx + d$$ and $$f(0), f(-1)$$ are odd integers, prove that $$f(x) = 0$$ cannot have all integral roots.

45. If equation $$2x^3 + ax^2 + bx + 4 = 0$$ has three real roots $$(a, b > 0),$$ prove that $$a + b \ge 6(2^{\frac{1}{3}} + 4^{\frac{1}{3}})$$

46. Find the condition that $$a_1x^3 + b_1x^2 + c_1x + d_1 = 0$$ and $$a_2x^3 + b_2x^2 + c_2x + d_2 = 0$$ have a common pair of repeated roots.

47. Let $$\alpha$$ be a non-zero real root of the equation $$a_1x^2 + b_1x + c_1 = 0$$. Find the condition for $$\alpha$$ to be repeated root of the equation $$a_2x^3 + b_2x^2 + c_2x + d_2 = 0$$

48. If $$\alpha, \beta, \gamma$$ are real roots of the equation $$x^3 - ax^2 + bx - c = 0,$$ prove that the area of the triangle whose sides are $$\alpha, \beta, \gamma$$ is $$\frac{1}{4}\sqrt{a(4ab - a^3 - 8c)}.$$

49. If $$a < b < c < d,$$ then show that the quadratic equation $$\mu(x - a)(x - c) + \lambda(x - b)(x - d) = 0$$ has real roots for all real $$\mu$$ and $$\lambda.$$

50. Show that equation $$3x^5 - 5x^3 + 21x + 3\sin x + 4\cos x + 5 = 0$$ can have at most one real root.