# 39. Quadratic Equations Problems Part 1¶

1. For what values of $$(1 + m)x^2 - 2(1 + 3m)x + (1 + 8m) = 0$$ has equal roots?

2. If $$a + b + c = 0$$ and $$a, b, c$$ are rational. Prove that the roots of the equation $$(b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0$$ are rational.

3. Show that if the roots of the equation $$(a^2 + b^2)x^2 + 2x(ac + bd) + c^2 + d^2 = 0$$ are real, they will be equal.

4. If the roots of the equation $$a(b - c)x^2 + b(c - a)x + c(a - b) = 0$$ be equal, prove that $$a, b, c$$ are in H. P.

5. If $$a + b + c = 0$$ and $$a, b, c$$ are real, then prove that equation $$(b - x)^2 - 4(a - x)(c - x) = 0$$ has real roots and roots will not be equal unless $$a = b = c.$$

6. Show that if $$p, q, r, s$$ are real numbers and $$pr = 2(q + s)$$ then at least one of the equations $$x^2 + px + q = 0$$ and $$x^2 + rx + s = 0$$ has real roots.

7. If the equation $$x^2 - 2px + q = 0$$ has two equal roots, then the equation $$(1 + y)x^2 - 2(p + y)x + (q + y) = 0$$ will have its roots real and distinct only when $$y$$ is negative and $$p$$ is not unity.

8. If the equation $$ax^2 + 2bx + c = 0$$ has real roots $$a, b, c$$ being real numbers and if $$m$$ and $$n$$ are real numbers such that $$m^2 > n^2 > 0$$ then prove that the equation $$ax^2 + 2mbx + nc = 0$$ has real roots.

9. If the equations $$ax + by = 1$$ and $$cx^2 + dy^2 = 1$$ have only one solution, prove that $$\frac{a^2}{c} + \frac{b^2}{d} = 1$$ and $$x = \frac{a}{c}, y = \frac{b}{d}$$

10. If $$r$$ be the ratio of the roots of the equation $$ax^2 + bx + c = 0,$$ show that $$\frac{(r + 1)^2}{r} = \frac{b^2}{ac}$$

11. If one root of the eq. $$(l - m)x^2 + lx + 1 = 0$$ be double of the other and if $$l$$ be real, show that $$m \leq \frac{9}{7}.$$

12. If one root of the quadratic equation $$ax^2 + bx + c = 0$$ is equal to the nth power of the other, then show that

$(ac^n)^{\frac{1}{n + 1}} + (a^nc)^{\frac{1}{n + 1}} + b = 0$
13. If the roots of the equation $$ax^2 + cx + c = 0$$ be in the ratio $$p:q,$$ show that

$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0$
14. If $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + px + q = 0$$. Find the value of the following in terms of $$p$$ and $$q$$.

1. $$\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}$$

2. $$(\omega\alpha + \omega^2\beta)(\omega^2\alpha + \omega\beta),$$ where $$\omega$$ an imaginary cube root of unity.

15. If $$\alpha$$ and $$\beta$$ be the roots of the equation $$A(x^2 + m^2) + Amx + cm^2x^2 = 0,$$ prove that $$A(\alpha^2 + \beta^2) + A\alpha\beta + c\alpha^2\beta^2 = 0$$

16. If $$\alpha$$ and $$\beta$$ be the roots of the equation $$ax^2 + bx + c = 0,$$ prove that $$a\left(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\right) + b\left(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right) = b$$

17. If $$a$$ and $$b$$ are the roots of the equation $$x^2 + px + 1 = 0$$ and $$c, d$$ are the roots of the equation $$x^2 + qx + 1 = 0,$$ show that $$q^2 - p^2 = (a - c)(b - c)(a + d)(b + d)$$

18. If the roots of the equation $$x^2 + px + q = 0$$ differ from the roots of the equation $$x^2 + qx + p = 0$$ by the same quantity, show that $$p + q + 4 = 0.$$

19. If $$\alpha, \beta$$ are the roots of the equation $$ax^2 + bx + c = 0$$ and $$S_n = \alpha^2 + \beta^n,$$ show that $$aS_{n + 1} + bS_n + cS_{n - 1} = 0$$ and hence find $$S_5.$$

20. If the sum of roots of the equation $$ax^2 + nx + c = 0$$ is equal to the sum of the squares of their reciprocals, show that $$bc^2, ca^2, ab^2$$ are in A. P.

21. If $$\alpha$$ and $$\beta$$ be the values of $$x$$ obtained from the equation $$m^2(x^2 - x) + 2mx + 3 = 0$$ and if $$m_1$$ and $$m_2$$ be the two values of $$m$$ for which $$\alpha$$ and $$\beta$$ are connected by the relation $$\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{4}{3},$$ find the value of $$\frac{m_1^2}{m_2} + \frac{m_2^2}{m_1}$$

22. If the ratio of the roots of the equation $$ax^2 + bx + c = 0$$ be equal to the roots of equation $$a_1x^2 + b_1x + c_1 = 0,$$ prove that $$\left(\frac{b}{b_1}\right)^2 = \frac{ca}{c_1a_1}$$

23. Find the quadratic equation with rational coefficients one of whose roots is $$\frac{1}{2 + \sqrt{5}}$$.

24. If $$\alpha$$ and $$\beta$$ be the roots of the equation $$ax^2 + bx + c =0$$ find the quadratic equation whose roots are $$\frac{1}{a\alpha + b}$$ and $$\frac{1}{a\beta + b}$$.

25. If $$c, d$$ are the roots of the equation $$(x - a)(x - b) - k = 0$$ show that $$a, b$$ are the roots of the equation $$(x - c)(x - d) + k = 0$$.

26. The coefficient of $$x$$ in the equation $$x^2 + px + q = 0$$ was wrongly written as $$17$$ in place of $$13$$ and roots found to be were $$-2$$ and $$-15$$. Find the roots of the correct equation.

27. If $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + px + q = 0,$$ show that $$\frac{\alpha}{\beta}$$ is a root of the equation $$qx^2 - (p^2 - 2q)x + q = 0.$$

28. If $$x^2 - ax + b = 0$$ and $$x^2 - px + q = 0$$ have a common root and the second equation has equal roots, show that $$b + q = \frac{ap}{2}$$

29. If $$ax^2 + 2bx + c = 0$$ and $$a_1x^2 + 2b_1x + c_1 = 0$$ have a common root and $$\frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1}$$ are in A. P., show that $$a_1, b_1, c_1$$ are in G. P.

30. If each pair of the following three equations $$x^2 + p_1x + q_1 = 0, x^2 + p_2x +q_2 = 0, x^2 + p_3x + q_3 = 0$$ has exactly one root in common, then show that $$(p_1 + p_2 + p_3)^2 = 4(p_1p_2 + p_2p_3 + p_3p_1 - q_1 - q_2 - q_3)$$

31. If the equations $$x^2 + cx + cb = 0$$ and $$x^2 + bx + ca = 0$$ have a common root, show that $$a + b + c = 0;$$ show that their other roots are given by the equation $$x^2 + ax + bc = 0$$

32. If $$a, b, c \in R$$ and equations $$ax^2 + bx + c = 0$$ and $$x^2 + 2x + 9 = 0$$ have a common root, show that $$a : b : c = 1 : 2 : 9$$

33. Find the value of $$p$$ if the equation $$3x^2 - 2x + p = 0$$ and $$6x^2 - 17x + 12 = 0$$ have a common root.

34. Show that $$|x|^2 - |x| - 2 = 0$$ is an equation.

35. Show that $$\frac{(x + b)(x + c)}{(b - a)(c - a)} + \frac{(x + c)(x + a)}{(c - b)(a - b)} + \frac{(x + a)(x + b)}{(a - c)(b - c)} = 1$$ is an identity.

36. If $$a, b, c, a_1, b_1, c_1$$ are rational and equations $$ax^2 + 2bx + c = 0$$ and $$a_1x^2 + 2b_1x + c_1 = 0$$ have one and only one root in common, prove that $$b^2 - ac$$ and $$b_1^2 - a_1c_1$$ must be perfect squares.

37. If $$(a^2 - 1)x^2 + (a - 1)x + a^2 - 4a + 3 = 0$$ be an identity in $$x,$$ then find the value of $$a$$.

38. Solve: $$\left(x + \frac{1}{x}\right)^2 = 4 + \frac{3}{2}\left(x + \frac{1}{x}\right)$$

39. Solve the equation: $$(x + 4)(x + 7)(x + 8)(x + 11) + 20 = 0$$

40. Solve the equation $$3^{2x + 1} + 3^2 = 3^{x + 3} + 3^x.$$

41. Solve for $$x:$$ $$(5 + 2\sqrt{6})^{x^2 - 3} + (5 - 2\sqrt{6})^{x^2 - 3} = 10$$

42. A car travels $$25$$ km per hour faster than a bus for a journey of $$500$$ km. The bus takes 10 hours more than the car. Find the speed of the bus and the car.

43. Show that the roots of the equation $$(a + b)^2x^2 - 2(a^2 - b^2)x + (a - b)^2 = 0$$ are equal.

44. Show that the equation $$3x^2 + 7x + 8 = 0$$ cannot be satisfied by any real value of $$x$$.

45. For what value of $$a$$ will the roots of the equation $$3x^2 + (7 + a) + 8 - a = 0$$ be equal.

46. If the roots of the equation $$(a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0$$ are equal, then show that $$a : b = c : d$$

47. Prove that the roots of the equation $$(b - c)x^2 + 2(c - a)x + (a - b) = 0$$ are always real.

48. Show that the roots of the equation $$\frac{1}{x - a} + \frac{1}{a} + \frac{1}{x - 1} = 0$$ are real for all real values of $$a$$.

49. Show that if $$a + b + c = 0,$$ the roots of the equation $$ax^2 + bx + c = 0$$ are rational.

50. Prove that the roots of the equation $$(b + c -2a)x^2 + (c + a - 2b)x + (a + b - 2c) = 0$$ are rational.