.. meta::
   :author: Shiv Shankar Dayal
   :title: Quadratic Equations Problems Part 6
   :description: Quadratic Equations Problems Part 6
   :keywords: quadratic equations, algebra

Quadratic Equations Problems Part 6
***********************************
251. If :math:`a, b, c` are rational, show that the roots of equation :math:`abc^2x^2 + 3a^2cx + b^2cx - 6a^2 - ab + 2b^2
     = 0` are rational.
252. If the roots of the equation :math:`ax^2 + bx + c = 0` be in the ration :math:`m:n` prove that
     :math:`\sqrt{\frac{m}{n}} + \sqrt{\frac{n}{m}} + \frac{b}{\sqrt{ac}} = 0`
253. If one root of the equation :math:`x^2 + x.f(a) + a = 0` is equal to the third power of the other, determine the
     function :math:`f(x)`.
254. If :math:`\alpha, \beta` are the roots of the equation :math:`x^2 - px + q = 0,` then find the quadratic equation
     the roots of which are :math:`(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)` and :math:`\alpha^3\beta^2 +
     \alpha^2\beta^3`
255. If :math:`\alpha, \beta` are the roots of the equation :math:`x^2 - bx + c = 0,` then find the quadratic equation
     whose roots are :math:`(\alpha^2 + \beta^2)(\alpha^3 + \beta^3)` and :math:`\alpha^5\beta^3 + \alpha^3\beta^5 -
     2\alpha^4\beta^4`.
256. If the sum of the roots of the quadratic equation :math:`ax^2 + bx + c = 0` is equal to the sum of the squares of
     their reciprocals, then show that :math:`\frac{b^2}{ac} + \frac{bc}{a^2} = 2.`
257. The time of oscillation of a rigid body about a horizontal axis at a distance :math:`h` from the C. G. is given by
     :math:`T = 2\pi \sqrt{\frac{h^2 + k^2}{gh}},` where :math:`k` is a constant. Show that there are two values for
     :math:`h` for a given value of :math:`T`. If :math:`h_1` and :math:`h_2` are two values of :math:`h,` show that
     :math:`h_1 + h_2 = \frac{gt^2}{4\pi^2}` and :math:`h_1h_2 = k^2.`
258. If :math:`\alpha_1,\alpha_2` be the roots of the equation :math:`x^2 + px + q = 0` and :math:`\beta_1,\beta_2` be
     roots of :math:`x^2 + rx + s = 0` and the system of equations :math:`\alpha_1y + \alpha_2z = 0` and :math:`\beta_1y
     + \beta_2z = 0` has non trivial solutions then show that :math:`\frac{p^2}{r^2} = \frac{q}{s}`
259. If :math:`a, b, c` are in H. P. and :math:`\alpha, \beta` be the roots of :math:`ax^2 + bx + c = 0,` show that
     :math:`-(1 + \alpha\beta)` is the H. M. of :math:`\alpha` and :math:`\beta.`
260. If :math:`\alpha, \beta` are the roots of the equation :math:`x + 1 = \lambda x(1 - \lambda x)` and if
     :math:`\lambda_1, lambda_2` are the two values of :math:`\lambda` determined from the equation
     :math:`\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = r - 2,` show that :math:`\frac{\lambda_1^2}{\lambda_2^2} +
     \frac{\lambda_2^2}{\lambda_1^2} + 2 = 4\left(\frac{r + 1}{r - 1}\right)^2`
261. If the roots of equation :math:`ax^2 + bx + c = 0` are reciprocals of those of :math:`lx^2 + mx + n = 0,` then
     prove that :math:`a:b:c = n:m:l,` where :math:`a, b, c, l, m, n` are all non zero.
262. If :math:`x_1, x_2` be the roots of the equation :math:`x^2 - 3x + A = 0` and :math:`x_3, x_4` be those of the
     equation :math:`x^2 - 12x + B = 0` and :math:`x_1, x_2, x_3, x_4` be an increasing G. P. find :math:`A` and
     :math:`B`.
263. Let :math:`p` and :math:`q` be roots of the equation :math:`x^2 - 2x + A = 0` and let :math:`r` and :math:`s` be
     the roots of the equation :math:`x^2 - 18x + B = 0.` If :math:`p < q < r < s` are in A. P. find the values of
     :math:`A` and :math:`B.`
264. Let :math:`\alpha, \beta` be the roots of the equation :math:`x^2 + ax - \frac{1}{2a^2} = 0, a` being a real
     parameter, prove that :math:`\alpha^4 + \beta^4 \ge 2 + \sqrt{2}.`
265. If :math:`\alpha, \beta` be the roots of the equation :math:`x^2 - px + q = 0` and :math:`\alpha > 0, \beta > 0,`
     then find the value of :math:`\alpha^{\frac{1}{4}} + \beta^{\frac{1}{4}}.`
266. If the difference between roots of the equation :math:`ax^2 - bx + c = 0` is same as the difference between the
     roots of equation :math:`bx^2 - cx + a = 0,` then show that :math:`b^4 - a^2c^2 = 4ab(bc - a^2).`
267. If :math:`f(x) = 0` is a cubic equation with real roots :math:`\alpha, \beta, \gamma` in order of magnitudes, show
     that one root of equation :math:`f'(x) = 0` lies between :math:`\frac{1}{2}(\alpha + \beta)` and
     :math:`\frac{1}{2}(2\alpha + \beta)` and the other root lies between :math:`\frac{1}{2}(\beta + \gamma)` and
     :math:`\frac{1}{3}(2\beta + \gamma).`
268. Let :math:`D_1` be the discriminant and :math:`\alpha, \beta` be the roots of the equation :math:`ax^2 + bx + c =
     0` and :math:`D_2` be the discriminant and :math:`\gamma, \delta` be the roots of the equation :math:`px^2 + qx + r
     = 0.` If :math:`\alpha, \beta, \gamma, \delta` are in A. P. then prove that :math:`D_1:D_2 = a^2:p^2.`
269. If :math:`\alpha, \beta` be the roots of the equation :math:`ax^2 + bx + c = 0` and :math:`\alpha + h, \beta + h`
     be those of equation :math:`px^2 + qx + r = 0,` then show that :math:`\frac{b^2 - 4ac}{a^2} = \frac{q^2 - 4pr}{p^2}.`
270. If :math:`\alpha, \beta` be the roots of the equation :math:`ax^2 + bx + c = 0` and :math:`\alpha + h, \beta + h`
     be those of equation :math:`px^2 + qx + r = 0,` then show that :math:`2h = \frac{b}{a} - \frac{q}{p}`
271. If :math:`\alpha, \beta` be the real and distinct roots of equation :math:`ax^2 + bx + c = 0` and :math:`\alpha^4,
     \beta^4` be those of equation :math:`lx^2 + mx + n = 0,` prove that the roots of equation :math:`a^2lx^2 - 4aclx +
     2c^2l + a^2m = 0` are real and opposite in sign.
272. If :math:`\alpha, \beta` be the roots of equation :math:`ax^2 + bx + c = 0` and :math:`\gamma, \delta` those of
     equation :math:`lx^2 + mx + n = 0,` then find the equation whose roots are :math:`\alpha\gamma + \beta\delta` and
     :math:`\alpha\delta + \beta\gamma.`
273. If :math:`p, q` be the roots of the equation :math:`x^2 + bx + c = 0,` prove that :math:`b`
     and :math:`c` are the roots of the equation :math:`x^2 + (p + q - pq)x - pq(p + q) = 0.`
274. If :math:`3p^2 = 5p + 2` and :math:`3q^2 = 5q + 2` where :math:`p \ne q,` obtain the equation whose roots are
     :math:`3p - 2q` and :math:`3q - 2p.`
275. If :math:`\alpha \pm \sqrt{\beta}` be the roots of the equation :math:`x^2 + px + q = 0,` prove that
     :math:`\frac{1}{\alpha} \pm \frac{1}{\sqrt{\beta}}` will be the roots of the equation :math:`(p^2 - 4q)(p^2x^2 +
     4px) = 16q`
276. If :math:`\alpha, \beta` be the roots of the equation :math:`x^2 - px + q = 0,` form the equation whose roots are
     :math:`\alpha^2\left(\frac{\alpha^2}{\beta} - \beta\right)` and :math:`\beta^2\left(\frac{\beta^2}{\alpha} -
     \alpha\right)`
277. Let :math:`a, b, c, d` be real numbers in G. P. If :math:`u, v, w` satisfy the system of equations :math:`u + 2v +
     3w = 6, 4u + 5v + 6w = 12, 6u + 9v = 4` then show that the roots of the equation :math:`\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 :math:`20x^2 + 10(a
     - d)^2x - 9 = 0` are reciprocals of each other.
278. If :math:`\alpha_1, \alpha_2, ..., \alpha_n` be the roots of the equation :math:`(\beta_1 - x)(\beta_2 - x) ...
     (\beta_n - x) + A = 0,` find the equation whose roots are :math:`\beta_1, \beta_2, ..., \beta_n.`
279. If :math:`\alpha_1, \alpha_2, ..., \alpha_n` be the roots of the equation :math:`x^n + nax - b = 0,` show that
     :math:`(\alpha_1 - \alpha_2)(\alpha_1 - \alpha_3) ... (\alpha_1 - \alpha_n) = n(\alpha^{n - 1} + a)`
280. If :math:`\alpha, \beta, \gamma, \delta` be the real roots of equation :math:`x^4 + qx^2 + rx + t = 0,` find the
     quadratic equation whose roots are :math:`(1 + \alpha^2)(1 + \beta^2)(1 + \gamma^2)(1 + \delta^2)` and :math:`1.`
281. If :math:`\alpha, \beta, \gamma` be the roots of the equation :math:`x^3 + px + q = 0,` find the cubic equation
     whose roots are :math:`\frac{\alpha + 1}{\alpha}, \frac{\beta + 1}{\beta}, \frac{\gamma + 1}{\gamma}.`
282. Show that one of the roots of the equation :math:`ax^2 + bx + c = 0` may be reciprocal of one of the roots of
     :math:`a_1x^2 + b_1x + c_1 = 0` if :math:`(aa_1 - cc_1)^2 = (hc_1 - ab_1)(b_1c - a_1b)`
283. If every pair of the equations :math:`x^2 + px + qr = 0, x^2 + qx + pr = 0` and :math:`x^2 + rx + pq = 0` have a
     common root, find the sum of the three common roots.
284. If equation :math:`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
     :math:`4x^2\sin^2\theta - 4\sin\theta + 1 = 0` and they have a common root then, find the value of :math:`\theta.`
285. If :math:`a \ne 0,` find the value of :math:`a` for which one of the roots of equation :math:`x^2 - x + 3a = 0` is
     double the roots of the quadratic equation :math:`x^2 - x + a = 0.`
286. If by eliminating :math:`x` between the equations :math:`x^2 + ax + b = 0` and :math:`xy + l(x + y) + m = 0,` a
     quadratic equation in terms of :math:`y` is formed whose roots are same as those of original quadratic equation in
     :math:`x,` then prove that either :math:`a = 2l` or :math:`b = m` or :math:`b + m = al.`
287. The roots of equation :math:`10x^3 - cx^2 - 54x - 27 = 0` are in H. P., then find :math:`c.`
288. If :math:`a, b, c` are the roots of the equation :math:`x^3 + px^2 + qx + r = 0` such that :math:`c^2 = -ab,` show
     that :math:`(2q - p^2)^3.r = (pq - 4r)^3`
289. Let :math:`\alpha + i\beta, \alpha, \beta \in R` be roots of the equation :math:`x^3 + qx + r = 0, q, r \in R.`
     Find a real cubic equation independent of :math:`\alpha` and :math:`\beta,` whose one root is :math:`2\alpha.`
290. If :math:`\alpha, \beta, \gamma` be the roots of the equation :math:`2x^3 + x^2 - 7 = 0,` show that :math:`\sum
     \left(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right) = -3`
291. The equations :math:`x^3 + px^2 + qx + r = 0` and :math:`x^3 + p'x^2 + q'x + r' = 0` have two common roots, find
     the quadratic equations whose roots are these common roots.
292. Find the condition that the roots of equation :math:`ax^3 + 3bx^2 + 3cx + d = 0` may be in G. P.
293. Find the condition that the equation :math:`x^3 - px^2 + qx - r = 0` have its roots in H. P.
294. If :math:`f(x) = x^3 + bx^2 + cx + d` and :math:`f(0), f(-1)` are odd integers, prove that :math:`f(x) = 0` cannot
     have all integral roots.
295. If equation :math:`2x^3 + ax^2 + bx + 4 = 0` has three real roots :math:`(a, b > 0),` prove that :math:`a + b \ge
     6(2^{\frac{1}{3}} + 4^{\frac{1}{3}})`
296. Find the condition that :math:`a_1x^3 + b_1x^2 + c_1x + d_1 = 0` and :math:`a_2x^3 + b_2x^2 + c_2x + d_2 = 0` have
     a common pair of repeated roots.
297. Let :math:`\alpha` be a non-zero real root of the equation :math:`a_1x^2 + b_1x + c_1 = 0`. Find the condition for
     :math:`\alpha` to be repeated root of the equation :math:`a_2x^3 + b_2x^2 + c_2x + d_2 = 0`
298. If :math:`\alpha, \beta, \gamma` are real roots of the equation :math:`x^3 - ax^2 + bx - c = 0,` prove that the
     area of the triangle whose sides are :math:`\alpha, \beta, \gamma` is :math:`\frac{1}{4}\sqrt{a(4ab - a^3 - 8c)}.`
299. If :math:`a < b < c < d,` then show that the quadratic equation :math:`\mu(x - a)(x - c) + \lambda(x - b)(x - d) =
     0` has real roots for all real :math:`\mu` and :math:`\lambda.`
300. Show that equation :math:`3x^5 - 5x^3 + 21x + 3\sin x + 4\cos x + 5 = 0` can have at most one real root.