39. Quadratic Equations Problems Part 1ΒΆ

  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.