.. meta::
   :author: Shiv Shankar Dayal
   :title: Quadratic Equations Problems Part 5
   :description: Quadratic Equations Problems Part 5
   :keywords: quadratic equations, algebra

Quadratic Equations Problems Part 5
***********************************
201. If roots of the equation :math:`ax^2 + bx + c = 0` be :math:`\frac{k + 1}{k}` and :math:`\frac{k + 2}{k + 1},`
     prove that :math:`(a + b + c)^2 = b^2 - 4ac.`
202. If :math:`f(x) = ax^2 + bx + c,` and :math:`\alpha, \beta` be the roots of the equation :math:`px^2 + qx + r = 0,`
     show that :math:`f(\alpha)f(\beta) = \frac{(cp - ar)^2 - (bp - aq)(cq - br)}{p^2}`. Hence or otherwise, show that
     if :math:`ax^2 + bx + c = 0` and :math:`px^2 + qx + r = 0` have a common root, then :math:`bp -aq, cp - ar` and
     :math:`cq - br` are in G. P.
203. If :math:`a(p + q)^2 + 2pbq + c = 0` and :math:`a(p + r)^2 + 2bpr + c = 0,` then show that :math:`qr = p^2 +
     \frac{c}{a}.`
204. If :math:`\alpha, \beta` are the roots of the equation :math:`x^2 - p(x + 1) - c = 0,` show that :math:`(\alpha +
     1)(\beta + 1) = 1 - c.` Hence, prove that :math:`\frac{\alpha^2 + 2\alpha + 1}{\alpha^2 + 2\alpha + c} +
     \frac{\beta^2 + 2\beta + 1}{\beta^2 + 2\beta + c} = 1`
205. If :math:`\alpha, \beta` be the roots of the equation :math:`x^2 + px + q = 0` and :math:`x^{2n} + p^nx^n + q^n =
     0,` where :math:`n` is an even integer, prove that :math:`\frac{\alpha}{\beta}, \frac{\beta}{\alpha}` are the roots
     of the equation :math:`x^2 + 1 + (x + 1)^n = 0.`
206. If :math:`\alpha, \beta` are the roots of the equation :math:`x^2 + px + q = 0` and also of equation
     :math:`x^{2n} + p^nx^n + q^n = 0` and if :math:`\frac{\alpha}{\beta}, \frac{\beta}{\alpha}` are the roots of the
     equation :math:`x^n + 1 + (x + 1)^n = 0,` then prove that :math:`n` must be an even integer.
207. If the roots of the equation :math:`x^2 - ax + b = 0` be real and differ by less than :math:`c,` then show that
     :math:`b` must lie between :math:`\frac{a^2 - c^2}{4}` and :math:`\frac{a^2}{4}`
208. Let :math:`a, b` and :math:`c` be integers with :math:`a > 1,` and let :math:`p` be a prime number. Show that if
     :math:`ax^2 + bx + c` is equal to :math:`p` for two distinct integral values of :math:`x,` then it cannot be equal
     to :math:`2p` for any integral value of :math:`x`.
209. If :math:`\alpha` and :math:`\beta` are the roots of equation :math:`x^2 + px + q = 0` and :math:`\alpha^4,
     \beta^4` are the roots of the equation :math:`x^2 - rx + s = 0,` show that the equation :math:`x^2 - 4qx + 2q^2 - r
     = 0` has real roots.
210. If :math:`\alpha, \beta` are the roots of the equation :math:`ax^2 + bx + c = 0,` and :math:`\alpha_1, -\beta` are
     those of equation :math:`a_1x^2 + b_1x + c_1 = 0,` show that :math:`\alpha, \alpha_1` are the roots of the equation

     .. math::

        \frac{x^2}{\frac{b}{a} + \frac{b_1}{a_1}} + x + \frac{1}{\frac{b}{c} + \frac{b_1}{c_1}} = 0
211. How many quadratic equations are possible which remains unchanged when its roots are squared?
212. If :math:`a, b, c` are in G. P. then show that the equations :math:`ax^2 + 2bx + c = 0` and :math:`dx^2 + 2ex + f =
     0` have a common root if :math:`\frac{a}{d}, \frac{b}{e}, \frac{c}{f}` are in H. P.
213. If the three equations :math:`x^2 + ax + 12 = 0, x^2 + bx + 15 = 0` and :math:`x^2 + (a + b)x + 36 = 0` have a
     common positive root, find :math:`a, b` and the roots of the equation.
214. If :math:`m(ax^2 + 2bx + c) + px^2 + 2qx + r` can be expressed in the form of :math:`n(x + k)^2,` then show that
     :math:`(ak - b)(qk - r) = (pk - q)(bk - c).`
215. The real numbers :math:`x_1, x_2, x_3` satisfying the equation :math:`x^3 - x^2 + \beta x + \gamma = 0` are
     in A. P. Find the intervals in which :math:`\beta` and :math:`\gamma` must lie.
216. If equations :math:`x^3 + 3px^2 + 3qx + r = 0` and :math:`x^2 + 2px + q = 0` have a common root, show that
     :math:`4(p^2 - q)(q^2 - pr) = (pq - r)^2`
217. If :math:`c \ne 0` and the equations :math:`x^3 + 2ax^2 + 3bx + c = 0` and :math:`x^3 + ax^2 + 2bx = 0` have a
     common root, show that :math:`(c - 2ab)^2 = (2b^2 - ac)(a^2 - b)`
218. If equation :math:`x^3 + ax + b = 0` have only real roots, then prove that :math:`4a^3 + 27b^2 \le 0.`
219. Let :math:`\alpha` be a root of :math:`ax^2 + bc + c = 0` and :math:`\beta` be a root of :math:`-ax^2 + bx + c =
     0.` Show that there exists a root of the equation :math:`\frac{a}{2}x^2 + bx + c = 0` that lie between
     :math:`\alpha` and :math:\beta` or :math:`\beta` and :math:`\alpha` as the case may be(:math:`\alpha, \beta \ne
     0`).
220. If :math:`a, b, c \in R, a \ne 0` and the quadratic equation :math:`ax^2 + bx + c = 0` has no real root, then show
     that :math:`(a + b + c)c > 0.`
221. If :math:`a < b < c < d,` then show that the quadratic equation :math:`(x - a)(x - c) + \lambda (x - b)(x - d) = 0`
     has real roots for all real values of :math:`\lambda`.
222. If :math:`2a + 3b + 6c = 0, (a, b, c \in R)` then show that the equation :math:`ax^2 + bx + c = 0` has at least one
     root between :math:`0` and :math:`2.`
223. If :math:`a, b, c` be non-zero real numbers such that :math:`\int_0^1 (1 + \cos^8 x)(ax^2 + bx + c)dx = \int_0^2 (1
     + \cos^8 x)(ax^2 + bx + c)dx` show that equation :math:`ax^2 + bx + c = 0` has at least on real root between
     :math:`1` and :math:`2`.
224. Let :math:`f(x) = ax^2 + bx + c,` where :math:`a, b, c \in R` and :math:`a \ne 0.` If :math:`f(x) = x` has non-real
     roots, show that equation :math:`f(f(x)) = x` has all non-real roots.
225. Let :math:`a, b, c` be positive integers and consider all the quadratic equations of the form :math:`ax^2 - bx + c
     = 0` which have two distinct real roots in :math:`]0, 1[`. Find the least positive integers :math:`a` and :math:`b`
     for which such a quadratic equation exist.
226. If equation :math:`ax^2 - bx + c = 0` have two distinct real roots in :math:`(0, 1), a, b, c \in N,` then prove
     that :math:`\log_5(abc)\ge 2`
227. If equation :math:`ax^2 + bx + 6 = 0` does not have two distinct real roots, then find the least value of :math:`3a
     + b`.
228. If equation :math:`2x^3 + ax^2 + bx + 4 = 0` has :math:`3` real roots, where :math:`a, b > 0,` show that :math:`a +
     b > -6`.
229. Show that equation :math:`x^3 + 2x^2 + x + 5 = 0` has only one real root :math:`\alpha` such that :math:`[\alpha] =
     -3,` where :math:`[x]` denotes the integral part of :math:`x.`
230. Solve :math:`(x^2 + 2)^2 + 8x^2 = 6x(x^2 + 2)`
231. Solve :math:`3x^3 = (x^2 + \sqrt{18}x + \sqrt{32})(x^2 - \sqrt{18}x - \sqrt{32}) - 4x^2`
232. Solve :math:`(15 + 4\sqrt{14})^t + (15 - 4\sqrt{14})^t = 30` where :math:`t = x^2 - 2|x|`
233. For :math:`a \le 0,` determine all the roots of the equation :math:`x^2 - 2a|x - a| - 3a^2 = 0`
234. Find all solutions of equation :math:`|x^2 - x - 6| = x + 2,` where :math:`x` is a real number.
235. Solve the equation :math:`2^{|x + 2|} - |2^{x + 1} - 1| = 2^{x + 1} + 1`
236. Solve :math:`3^x + 4^x + 5^x = 6^x`
237. Solve :math:`(\sqrt{2 + \sqrt{3}})^x + (\sqrt{2 - \sqrt{3}})^x = 2^x`
238. Let :math:`\{x\}` and :math:`[x]` denote the fractional and integral part of a real number :math:`x`
     respectively. Solve :math:`4\{x\} = x + [x]`
239. For the same notation as previous problem solve :math:`[x]^2 = x(x - [x])`
240. Solve :math:`x^3 - y^3 = 127, x^2y - xy^2 = 42`
241. Solve the system of equations: :math:`x - 2y + z = 0, 4x - y - 3z = 0, x^2 - 2xy + 3xz = 14`
242. Solve :math:`x^4 + y^4 = 82, x+ y = 4`
243. Solve :math:`\sqrt{a(2^x - 2) + 1} = 1 - 2^x, x \in R`
244. If :math:`x` is an integer, find the integral values of :math:`m` satisfying the equation :math:`(x - 5)(x + m) + 2
     = 0`
245. Find the positive solutions of the system of equations :math:`x^{x + y} = y^n` and :math:`y^{x + y}= x^{2n}y^n`
     where :math:`n > 0`
246. Solve the equation :math:`(144)^{|x|} - 2(12)^{|x|} + a = 0` for every value of the parameter :math:`a.`
247. If :math:`m` and :math:`n` are odd integers show that the equation :math:`x^2 + 2mx + 2n = 0` cannot have rational
     roots.
248. If :math:`f(x) = ax^3 + bx^2 + cx + d` has local extrema at two points of opposite sign, then prove that the roots
     of the quadratic equation :math:`ax^2 + bx + c = 0` are real and distinct.
249. If :math:`a` and :math:`b` are real, :math:`b\ne 0,` prove that the roots of the quadratic equation :math:`\frac{(x
     - a)(ax - 1)}{x^2 - 1} = b,` can never be equal.
250. If :math:`n` and :math:`r` are positive integers such that :math:`0 < r < n,` then show that the roots of the
     quadratic equation :math:`^nC_rx^2 + 2~^nC_{r + 1}x + ^nC_{r + 2} = 0` are real.