48. Quadratic Equations Problems Part 2

51. Show that the roots of the equation \(x^2 + rx + s = 0\) will be rational if \(r = k + \frac{s}{k},\) where \(r, s\) and \(k\) are rational.

52. Prove that roots of the equation \((x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0\) are always real and cannot be equal unless \(a = b = c\)

53. If \(a, b, c\) are rational, show that the roots of the equation \(a^2(b^2 - c^2)x^2 + b^2(c^2 - a^2)x + c^2(a^2 - b^2) = 0\) are rational.

54. Show that the roots of the equation \((a^4 + b^4)x^2 + 4abcdx + c^4 + d^4 = 0\) cannot be different if real.

55. If \(p, q, r\) are in H. P. and \(p\) and \(r\) are of the same sign; prove that the roots of the equation \(px^2 + 2qx + r = 0\) will be complex.

56. Prove that the roots of the equation \(bx^2 + (b - c)x + (b - c - a) = 0\) are real if those of equation \(ax^2 + 2bx + b = 0\) are imaginary and vice-versa.

57. Prove that the values of \(x\) obtained from the equation \(ax^2 + by^2 = 1\) and \(ax + by = 1\) will be equal if \(a + b = 1\)

58. Prove that the two values of \(x\) obtained from the equations \(x^2 + y^2 = a^2\) and \(y = mx + c\) will be equal if \(c^2 = a^2(1 + m^2)\)

59. The roots of the equation \(4x^2 - (5a + 1)x + 5a = 0\) are \(\alpha\) and \(\beta\). If \(\beta = 1 + \alpha,\) calculate the possible values of \(a, \alpha\) and \(\beta\).

60. If one root of the equation \(5x^2 + 13x + k = 0\) be reciprocal of another, find \(k.\)

61. Find the values of \(m,\) for which the equation \(5x^2 - 4x + 2 + m(4x^2 - 2x - 1) = 0\) has (a) equal roots (b) the products of roots is \(2\) (c) the sum of roots is \(6\).

62. Find the relation between the coefficients of the quadratic equation \(ax^2 + bx + c = 0\) if one root be \(n\) times another.

63. If the roots of the equation \(ax^2 + bx + c = 0\) bear to one another the ratio \(3:4,\) prove that \(12b^2 = 49ac\)

64. If the roots of the equation \(4x^2 + ax + 3 = 0\) are in the ratio \(1:2\) show that the roots of the equation \(ax^2 + 3x + a = 2\) are imaginary.

65. If one root of the equation \(x^2 - px + q = 0\) be \(m\) times their difference, prove that \(p^2(m^2 - 1) = 4m^2q\)

66. If the difference of the roots of \(x^2 - px + q = 0\) is unity, then prove that \(p^2 - 4q = 1\) and \(p^2 + 4q = (1 + 2q)^2\)

67. Find the condition that the equation \(\frac{a}{x - a} + \frac{b}{x - b} = m\) may have roots equal in magnitude but opposite in sign.

68. Find the relation between coefficients of the equation \(ax^2 + bx + c = 0\) if one roots exceeds other by \(k\).

69. If one root of the equation \(ax^2 + bx + c = 0\) be the square of the other, show that \(b^3 + a^2c + ac^2 = 3abc\)

70. Determine the value of \(p\) for which one root of the equation \(x^2 + px + 1 = 0\) is the square of the other.

71. If one root of the equation \(x^2 + px + q = 0\) be the square of the other then show that \(p^3 - q(3p - 1) + q^2 = 0\)

72. If \(\alpha, \beta\) be the roots of the equation \(2x^2 + 3x + 4 = 0.\) Find the values of

    1. \(\alpha^2 + \beta^2\)

    2. \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)

73. If \(\alpha, \beta\) are the roots of the equation \(ax^2 + bx + c = 0,\) find the values of the following in terms of \(a, b, c\)

    \(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\)

74. If \(\alpha, \beta\) be the roots of the equation \(ax^2 + bx + b = 0,\) prove that

    \(\sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} + \sqrt{\frac{b}{a}} = 0\)

75. Show that the two equations \(x^2 - 2ax + b^2 = 0\) and \(x^2 - 2bx + b^2 = 0\) are such that the G. M. of the roots of one is equal to the A. M. of the roots of the another.

76. If sum of roots of the equation \(px^2 + qx + r = 0\) be equal to the sum of their squares, show that \(2pr = pq + q^2\)

77. If \(\alpha, \beta\) be the roots of the equation \(x^2 - px + q = 0,\) prove that

    \(\frac{\alpha^2}{\beta^2} + \frac{\beta^2}{\alpha^2} = \frac{p^4}{q^2} - \frac{4p^2}{q} + 2\)

78. If \(\alpha, \beta\) be the roots of the equation \(ax^2 + bx + c = 0,\) find the value of \(\frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2}\)

79. If \(\alpha, \beta\) be the roots of the equation \(\lambda(x^2 - x) + x + 5 = 0\) and if \(\lambda_1\) and \(\lambda_2\) are the two values for which the roots \(\alpha, \beta\) are connected by the relation \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{4}{5},\) then prove that

    1. \(\frac{\lambda_1}{\lambda_2} + \frac{\lambda_2}{\lambda_1} = 254\)

    2. \(\frac{\lambda_1^2}{\lambda_2} + \frac{\lambda_2^2}{\lambda_1} = 4048\)

80. If \(\alpha, \beta\) be the roots of the equation \(x^2 + px + q = 0\) and \(\gamma, \delta\) be the roots of \(x^2 + rx + s = 0\) find the values of

    1. \((\alpha + \gamma)(\alpha + \delta)(\beta + \gamma)(\beta + \delta)\)

    2. \((\alpha - \gamma)(\beta - \delta) + (\beta - \gamma)(\alpha - \delta)\)

    3. \((\alpha - \gamma)^2 + (\beta - \gamma)^2 + (\alpha - \delta)^2 + (\beta - \delta)^2\)

81. If \(\alpha, \beta\) be the roots of the equation \(x^2 - px + q = 0\) and if \(\nu_n = \alpha^n + \beta^n,\) prove that \(\nu_{n + 1} = p\nu_{n - 1} - q\nu_{n - 1}\)

82. If \(\alpha, \beta\) be the roots of the equation \(x^2 + px - q = 0\) and \(\gamma, \delta\) those of equation \(x^2 + px + r = 0,\) prove that

    \((\alpha - \gamma)(\alpha -\delta) = (\beta - \gamma)(\beta - \delta) = q + r\)

83. If \(\alpha, \beta\) be the roots of the equation \(x^2 - 2px + q = 0\) and \(\gamma, \delta\) those of the equation \(x^2 - 2rx + s = 0\) and if

    1. \(\alpha\delta = \beta\gamma,\) prove that \(p^2s = r^2q\)

    2. \(\alpha, \beta, \gamma, \delta\) be in G. P., prove that \(p^2s = r^2q\)

    3. \(\alpha, \beta, \gamma, \delta\) be in A. P., prove that \(s - q = r^2 - p^2\)

84. If the roots of the equation \(ax^2 + 2bx + c = 0\) be \(\alpha\) and \(\beta,\) and those of the equation \(Ax^2 + 2Bx + C = 0\) be \(\alpha + k\) and \(\beta + k,\) prove that

    \(\frac{b^2 - ac}{B^2 - AC} =\left(\frac{a}{A}\right)^2\)

85. If the roots of the equation \(ax^2 + bx + c = 0\) be \(\alpha\) and \(\beta,\) and those of the equation \(Ax^2 + Bx + C = 0\) be \(\alpha + k\) and \(\beta + k,\) prove that

    \(\frac{b^2 - 4ac}{B^2 - 4AC} =\left(\frac{a}{A}\right)^2\)

86. If the roots of the equation \(x^2 + 2px + q = 0\) and \(x^2 + 2qx + p = 0\) differ by a constant then show that \(p + q + 1 = 0\)

87. If \(\alpha, \beta\) be the roots of the equation \(ax^2 + bx + c = 0\) then find the equations whose roots are

    1. \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\)

    2. \(\frac{\alpha^2}{\beta}\) and \(\frac{\beta^2}{\alpha}\)

    3. \((\alpha + \beta)^2\) and \((\alpha - \beta)^2\)

    4. \(\frac{1 - \alpha}{1 + \alpha}\) and \(\frac{1 - \beta}{1 + \beta}\)

    5. \(\frac{1}{(\alpha + \beta)^2}\) and \((\alpha - \beta)^2\)

88. Find those equations whose roots are (i) reciprocal of the roots of (ii) equal in magnitude but opposite in sign to the roots of the equation \(ax^2 + bx + c = 0\)

89. If \(\alpha, \beta\) be the roots of the equation \(x^2 + px + q = 0,\) find the value of (i) \(\alpha^2 + \beta^4\) (ii) \(\alpha^{-4} + \beta^{-4}\)

90. If \(\alpha, \beta\) be the roots of the equation \(x^2 - px + q = 0,\) find the equation whose roots are

    1. \(\frac{q}{p - \alpha}\) and \(\frac{q}{p - \beta}\)

    2. \(\alpha + \frac{1}{\beta}\) and \(\beta + \frac{1}{\alpha}\)

91. Find the values of \(p\) and \(q\) such that the equation \(x^2 + px + q = 0\) has \(5 + 3i\) as a root.

92. Form the quadratic equation whose one root is \(3 + 4i.\)

93. If one root of the equation \(4x^2 + 2x - 1 = 0\) be \(\alpha\) then prove that its second root is \(4\alpha^3 - 3\alpha.\)

94. If \(\alpha \ne \beta\) and \(\alpha^2 = 5\alpha - 3, \beta^2 = 5\beta - 3,\) form the quadratic equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}.\)

95. In copying a quadratic equation of the form \(x^2 + px + q = 0,\) the coefficient of \(x\) was wrongly written as \(-10\) in place of \(-11\) and the roots were found to be \(4\) and \(6.\) Find the roots of the correct equation.

96. In writing a quadratic equation of the form \(x^2 + px + q = 0,\) the constant term was wrongly written as \(-6\) in place of \(2\) and the roots were found to be \(6\) and \(-1.\) Find the correct equation.

97. Two candidates attempt to solve a quadratic equation of the form \(x^2 + px + q = 0.\) One starts with a wrong value of \(p\) and finds the roots to be \(2\) and \(6\). The other starts with a wrong value of \(q\) and find the roots to be \(2\) and \(-9.\) Find the correct roots.

98. If \(\alpha, \beta\) be the roots of the quadratic equation \(x^2 + px + q = 0\) and \(\alpha_1, \beta_1\) be the roots of the equation \(x^2 - px + q = 0.\) Form the quadratic equation whose roots are \(\frac{1}{\alpha_1\beta} + \frac{1}{\alpha\beta_1}\) and \(\frac{1}{\alpha\alpha_1} + \frac{1}{\beta\beta_1}\)

99. If \(2 + \sqrt{3}i\) is a root of the equation \(x^2 + px + q = 0,\) where \(p, q\) are real, then find them.

100. Form the equation whose one root is \(\frac{1}{2 + \sqrt{3}}\)