Quadratic Equations Problems Part 1
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1. For what values of :math:`(1 + m)x^2 - 2(1 + 3m)x + (1 + 8m) = 0` has equal roots?
2. If :math:`a + b + c = 0` and :math:`a, b, c` are rational. Prove that the
   roots of the equation :math:`(b + c - a)x^2 + (c + a - b)x + (a + b - c) =
   0` are rational.
3. Show that if the roots of the equation :math:`(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 :math:`a(b - c)x^2 + b(c - a)x + c(a - b) = 0` be equal, prove that :math:`a, b, c` are in H. P.
5. If :math:`a + b + c = 0` and :math:`a, b, c` are real, then prove that equation :math:`(b - x)^2 - 4(a - x)(c - x) = 0` has real
   roots and roots will not be equal unless :math:`a = b = c.`
6. Show that if :math:`p, q, r, s` are real numbers and :math:`pr = 2(q + s)` then at least one of the equations :math:`x^2 + px + q
   = 0` and :math:`x^2 + rx + s = 0` has real roots.
7. If the equation :math:`x^2 - 2px + q = 0` has two equal roots, then the equation :math:`(1 + y)x^2 - 2(p + y)x + (q + y) = 0`
   will have its roots real and distinct only when :math:`y` is negative and :math:`p` is not unity.
8. If the equation :math:`ax^2 + 2bx + c = 0` has real roots :math:`a, b, c` being real numbers and if :math:`m` and :math:`n` are
   real numbers such that :math:`m^2 > n^2 > 0` then prove that the equation :math:`ax^2 + 2mbx + nc = 0` has real roots.
9. If the equations :math:`ax + by = 1` and :math:`cx^2 + dy^2 = 1` have only one solution, prove that :math:`\frac{a^2}{c} +
   \frac{b^2}{d} = 1` and :math:`x = \frac{a}{c}, y = \frac{b}{d}`
10. If :math:`r` be the ratio of the roots of the equation :math:`ax^2 + bx + c = 0,` show that :math:`\frac{(r + 1)^2}{r} =
    \frac{b^2}{ac}`
11. If one root of the eq. :math:`(l - m)x^2 + lx + 1 = 0` be double of the other and if :math:`l` be real, show that :math:`m \leq
    \frac{9}{7}.`
12. If one root of the quadratic equation :math:`ax^2 + bx + c = 0` is equal to the nth power of the other, then show that

    .. math::

       (ac^n)^{\frac{1}{n + 1}} + (a^nc)^{\frac{1}{n + 1}} + b = 0

13. If the roots of the equation :math:`ax^2 + cx + c = 0` be in the ratio :math:`p:q,` show that

    .. math::

       \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0

14. If :math:`\alpha` and :math:`\beta` be the roots of the equation :math:`x^2 + px + q = 0`. Find the value of the following in
    terms of :math:`p` and :math:`q`.

    i. :math:`\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}`
    ii. :math:`(\omega\alpha + \omega^2\beta)(\omega^2\alpha + \omega\beta),` where :math:`\omega` an imaginary cube root of unity.

15. If :math:`\alpha` and :math:`\beta` be the roots of the equation :math:`A(x^2 + m^2) + Amx + cm^2x^2 = 0,` prove that
    :math:`A(\alpha^2 + \beta^2) + A\alpha\beta + c\alpha^2\beta^2 = 0`
16. If :math:`\alpha` and :math:`\beta` be the roots of the equation :math:`ax^2 + bx + c = 0,` prove that
    :math:`a\left(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\right) + b\left(\frac{\alpha}{\beta} +
    \frac{\beta}{\alpha}\right) = b`
17. If :math:`a` and :math:`b` are the roots of the equation :math:`x^2 + px + 1 = 0` and :math:`c, d` are the roots of the equation
    :math:`x^2 + qx + 1 = 0,` show that :math:`q^2 - p^2 = (a - c)(b - c)(a + d)(b + d)`
18. If the roots of the equation :math:`x^2 + px + q = 0` differ from the roots of the equation :math:`x^2 + qx + p = 0` by the same
    quantity, show that :math:`p + q + 4 = 0.`
19. If :math:`\alpha, \beta` are the roots of the equation :math:`ax^2 + bx + c = 0` and :math:`S_n = \alpha^2 + \beta^n,` show that
    :math:`aS_{n + 1} + bS_n + cS_{n - 1} = 0` and hence find :math:`S_5.`
20. If the sum of roots of the equation :math:`ax^2 + nx + c = 0` is equal to the sum of the squares of their reciprocals, show that
    :math:`bc^2, ca^2, ab^2` are in A. P.
21. If :math:`\alpha` and :math:`\beta` be the values of :math:`x` obtained from the equation :math:`m^2(x^2 - x) + 2mx + 3 = 0` and
    if :math:`m_1` and :math:`m_2` be the two values of :math:`m` for which :math:`\alpha` and :math:`\beta` are connected by the
    relation :math:`\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{4}{3},` find the value of :math:`\frac{m_1^2}{m_2} +
    \frac{m_2^2}{m_1}`
22. If the ratio of the roots of the equation :math:`ax^2 + bx + c = 0` be equal to the roots of equation :math:`a_1x^2 + b_1x + c_1
    = 0,` prove that :math:`\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 :math:`\frac{1}{2 + \sqrt{5}}`.
24. If :math:`\alpha` and :math:`\beta` be the roots of the equation :math:`ax^2 + bx + c =0` find the quadratic equation whose
    roots are :math:`\frac{1}{a\alpha + b}` and :math:`\frac{1}{a\beta + b}`.
25. If :math:`c, d` are the roots of the equation :math:`(x - a)(x - b) - k = 0` show that :math:`a, b` are the roots of the
    equation :math:`(x - c)(x - d) + k = 0`.
26. The coefficient of :math:`x` in the equation :math:`x^2 + px + q = 0` was wrongly written as :math:`17` in place of :math:`13`
    and roots found to be were :math:`-2` and :math:`-15`. Find the roots of the correct equation.
27. If :math:`\alpha` and :math:`\beta` be the roots of the equation :math:`x^2 + px + q = 0,` show that
    :math:`\frac{\alpha}{\beta}` is a root of the equation :math:`qx^2 - (p^2 - 2q)x + q = 0.`
28. If :math:`x^2 - ax + b = 0` and :math:`x^2 - px + q = 0` have a common root and the second equation has equal roots, show that
    :math:`b + q = \frac{ap}{2}`
29. If :math:`ax^2 + 2bx + c = 0` and :math:`a_1x^2 + 2b_1x + c_1 = 0` have a common root and :math:`\frac{a}{a_1}, \frac{b}{b_1},
    \frac{c}{c_1}` are in A. P., show that :math:`a_1, b_1, c_1` are in G. P.
30. If each pair of the following three equations :math:`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 :math:`(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 :math:`x^2 + cx + cb = 0` and :math:`x^2 + bx + ca = 0` have a common root, show that :math:`a + b + c = 0;`
    show that their other roots are given by the equation :math:`x^2 + ax + bc = 0`
32. If :math:`a, b, c \in R` and equations :math:`ax^2 + bx + c = 0` and :math:`x^2 + 2x + 9 = 0` have a common root, show that
    :math:`a : b : c = 1 : 2 : 9`
33. Find the value of :math:`p` if the equation :math:`3x^2 - 2x + p = 0` and :math:`6x^2 - 17x + 12 = 0` have a common root.
34. Show that :math:`|x|^2 - |x| - 2 = 0` is an equation.
35. Show that :math:`\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 :math:`a, b, c, a_1, b_1, c_1` are rational and equations :math:`ax^2 + 2bx + c = 0` and :math:`a_1x^2 + 2b_1x + c_1 = 0`
    have one and only one root in common, prove that :math:`b^2 - ac` and :math:`b_1^2 - a_1c_1` must be perfect squares.
37. If :math:`(a^2 - 1)x^2 + (a - 1)x + a^2 - 4a + 3 = 0` be an identity in :math:`x,` then find the value of :math:`a`.
38. Solve: :math:`\left(x + \frac{1}{x}\right)^2 = 4 + \frac{3}{2}\left(x + \frac{1}{x}\right)`
39. Solve the equation: :math:`(x + 4)(x + 7)(x + 8)(x + 11) + 20 = 0`
40. Solve the equation :math:`3^{2x + 1} + 3^2 = 3^{x + 3} + 3^x.`
41. Solve for :math:`x:` :math:`(5 + 2\sqrt{6})^{x^2 - 3} + (5 - 2\sqrt{6})^{x^2 - 3} = 10`
42. A car travels :math:`25` km per hour faster than a bus for a journey of :math:`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 :math:`(a + b)^2x^2 - 2(a^2 - b^2)x + (a - b)^2 = 0` are equal.
44. Show that the equation :math:`3x^2 + 7x + 8 = 0` cannot be satisfied by any real value of :math:`x`.
45. For what value of :math:`a` will the roots of the equation :math:`3x^2 + (7 + a) + 8 - a = 0` be equal.
46. If the roots of the equation :math:`(a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0` are equal, then show that :math:`a : b = c :
    d`
47. Prove that the roots of the equation :math:`(b - c)x^2 + 2(c - a)x + (a - b) = 0` are always real.
48. Show that the roots of the equation :math:`\frac{1}{x - a} + \frac{1}{a} + \frac{1}{x - 1} = 0` are real for all real values of
    :math:`a`.
49. Show that if :math:`a + b + c = 0,` the roots of the equation :math:`ax^2 + bx + c = 0` are rational.
50. Prove that the roots of the equation :math:`(b + c -2a)x^2 + (c + a - 2b)x + (a + b - 2c) = 0` are rational.