# 56. Quadratic Equations Problems Part 6

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.

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

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 \(f(x)\).

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 \(\alpha^3\beta^2 + \alpha^2\beta^3\)

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 \(\alpha^5\beta^3 + \alpha^3\beta^5 - 2\alpha^4\beta^4\).

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 \(\frac{b^2}{ac} + \frac{bc}{a^2} = 2.\)

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 \(h_1h_2 = k^2.\)

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 \(\frac{p^2}{r^2} = \frac{q}{s}\)

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 \(\beta.\)

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 \(\frac{\lambda_1^2}{\lambda_2^2} + \frac{\lambda_2^2}{\lambda_1^2} + 2 = 4\left(\frac{r + 1}{r - 1}\right)^2\)

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.

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 \(B\).

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 \(B.\)

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

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

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 \(b^4 - a^2c^2 = 4ab(bc - a^2).\)

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 \(\frac{1}{3}(2\beta + \gamma).\)

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 \(D_1:D_2 = a^2:p^2.\)

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 \(\frac{b^2 - 4ac}{a^2} = \frac{q^2 - 4pr}{p^2}.\)

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 \(2h = \frac{b}{a} - \frac{q}{p}\)

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.

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

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 \(x^2 + (p + q - pq)x - pq(p + q) = 0.\)

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

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 \((p^2 - 4q)(p^2x^2 + 4px) = 16q\)

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 \(\beta^2\left(\frac{\beta^2}{\alpha} - \alpha\right)\)

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.

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 \(\beta_1, \beta_2, ..., \beta_n.\)

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

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 \(1.\)

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

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 \((aa_1 - cc_1)^2 = (hc_1 - ab_1)(b_1c - a_1b)\)

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.

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 \(\theta.\)

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 \(x^2 - x + a = 0.\)

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 \(b + m = al.\)

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

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

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 \(2\alpha.\)

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

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.

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

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

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.

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

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.

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 \(a_2x^3 + b_2x^2 + c_2x + d_2 = 0\)

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 \(\frac{1}{4}\sqrt{a(4ab - a^3 - 8c)}.\)

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 \(\lambda.\)

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