# 54. Quadratic Equations Problems Part 5

If roots of the equation \(ax^2 + bx + c = 0\) be \(\frac{k + 1}{k}\) and \(\frac{k + 2}{k + 1},\) prove that \((a + b + c)^2 = b^2 - 4ac.\)

If \(f(x) = ax^2 + bx + c,\) and \(\alpha, \beta\) be the roots of the equation \(px^2 + qx + r = 0,\) show that \(f(\alpha)f(\beta) = \frac{(cp - ar)^2 - (bp - aq)(cq - br)}{p^2}\). Hence or otherwise, show that if \(ax^2 + bx + c = 0\) and \(px^2 + qx + r = 0\) have a common root, then \(bp -aq, cp - ar\) and \(cq - br\) are in G. P.

If \(a(p + q)^2 + 2pbq + c = 0\) and \(a(p + r)^2 + 2bpr + c = 0,\) then show that \(qr = p^2 + \frac{c}{a}.\)

If \(\alpha, \beta\) are the roots of the equation \(x^2 - p(x + 1) - c = 0,\) show that \((\alpha + 1)(\beta + 1) = 1 - c.\) Hence, prove that \(\frac{\alpha^2 + 2\alpha + 1}{\alpha^2 + 2\alpha + c} + \frac{\beta^2 + 2\beta + 1}{\beta^2 + 2\beta + c} = 1\)

If \(\alpha, \beta\) be the roots of the equation \(x^2 + px + q = 0\) and \(x^{2n} + p^nx^n + q^n = 0,\) where \(n\) is an even integer, prove that \(\frac{\alpha}{\beta}, \frac{\beta}{\alpha}\) are the roots of the equation \(x^2 + 1 + (x + 1)^n = 0.\)

If \(\alpha, \beta\) are the roots of the equation \(x^2 + px + q = 0\) and also of equation \(x^{2n} + p^nx^n + q^n = 0\) and if \(\frac{\alpha}{\beta}, \frac{\beta}{\alpha}\) are the roots of the equation \(x^n + 1 + (x + 1)^n = 0,\) then prove that \(n\) must be an even integer.

If the roots of the equation \(x^2 - ax + b = 0\) be real and differ by less than \(c,\) then show that \(b\) must lie between \(\frac{a^2 - c^2}{4}\) and \(\frac{a^2}{4}\)

Let \(a, b\) and \(c\) be integers with \(a > 1,\) and let \(p\) be a prime number. Show that if \(ax^2 + bx + c\) is equal to \(p\) for two distinct integral values of \(x,\) then it cannot be equal to \(2p\) for any integral value of \(x\).

If \(\alpha\) and \(\beta\) are the roots of equation \(x^2 + px + q = 0\) and \(\alpha^4, \beta^4\) are the roots of the equation \(x^2 - rx + s = 0,\) show that the equation \(x^2 - 4qx + 2q^2 - r = 0\) has real roots.

If \(\alpha, \beta\) are the roots of the equation \(ax^2 + bx + c = 0,\) and \(\alpha_1, -\beta\) are those of equation \(a_1x^2 + b_1x + c_1 = 0,\) show that \(\alpha, \alpha_1\) are the roots of the equation

\[\frac{x^2}{\frac{b}{a} + \frac{b_1}{a_1}} + x + \frac{1}{\frac{b}{c} + \frac{b_1}{c_1}} = 0\]How many quadratic equations are possible which remains unchanged when its roots are squared?

If \(a, b, c\) are in G. P. then show that the equations \(ax^2 + 2bx + c = 0\) and \(dx^2 + 2ex + f = 0\) have a common root if \(\frac{a}{d}, \frac{b}{e}, \frac{c}{f}\) are in H. P.

If the three equations \(x^2 + ax + 12 = 0, x^2 + bx + 15 = 0\) and \(x^2 + (a + b)x + 36 = 0\) have a common positive root, find \(a, b\) and the roots of the equation.

If \(m(ax^2 + 2bx + c) + px^2 + 2qx + r\) can be expressed in the form of \(n(x + k)^2,\) then show that \((ak - b)(qk - r) = (pk - q)(bk - c).\)

The real numbers \(x_1, x_2, x_3\) satisfying the equation \(x^3 - x^2 + \beta x + \gamma = 0\) are in A. P. Find the intervals in which \(\beta\) and \(\gamma\) must lie.

If equations \(x^3 + 3px^2 + 3qx + r = 0\) and \(x^2 + 2px + q = 0\) have a common root, show that \(4(p^2 - q)(q^2 - pr) = (pq - r)^2\)

If \(c \ne 0\) and the equations \(x^3 + 2ax^2 + 3bx + c = 0\) and \(x^3 + ax^2 + 2bx = 0\) have a common root, show that \((c - 2ab)^2 = (2b^2 - ac)(a^2 - b)\)

If equation \(x^3 + ax + b = 0\) have only real roots, then prove that \(4a^3 + 27b^2 \le 0.\)

Let \(\alpha\) be a root of \(ax^2 + bc + c = 0\) and \(\beta\) be a root of \(-ax^2 + bx + c = 0.\) Show that there exists a root of the equation \(\frac{a}{2}x^2 + bx + c = 0\) that lie between \(\alpha\) and :math:beta` or \(\beta\) and \(\alpha\) as the case may be(\(\alpha, \beta \ne 0\)).

If \(a, b, c \in R, a \ne 0\) and the quadratic equation \(ax^2 + bx + c = 0\) has no real root, then show that \((a + b + c)c > 0.\)

If \(a < b < c < d,\) then show that the quadratic equation \((x - a)(x - c) + \lambda (x - b)(x - d) = 0\) has real roots for all real values of \(\lambda\).

If \(2a + 3b + 6c = 0, (a, b, c \in R)\) then show that the equation \(ax^2 + bx + c = 0\) has at least one root between \(0\) and \(2.\)

If \(a, b, c\) be non-zero real numbers such that \(\int_0^1 (1 + \cos^8 x)(ax^2 + bx + c)dx = \int_0^2 (1 + \cos^8 x)(ax^2 + bx + c)dx\) show that equation \(ax^2 + bx + c = 0\) has at least on real root between \(1\) and \(2\).

Let \(f(x) = ax^2 + bx + c,\) where \(a, b, c \in R\) and \(a \ne 0.\) If \(f(x) = x\) has non-real roots, show that equation \(f(f(x)) = x\) has all non-real roots.

Let \(a, b, c\) be positive integers and consider all the quadratic equations of the form \(ax^2 - bx + c = 0\) which have two distinct real roots in \(]0, 1[\). Find the least positive integers \(a\) and \(b\) for which such a quadratic equation exist.

If equation \(ax^2 - bx + c = 0\) have two distinct real roots in \((0, 1), a, b, c \in N,\) then prove that \(\log_5(abc)\ge 2\)

If equation \(ax^2 + bx + 6 = 0\) does not have two distinct real roots, then find the least value of \(3a + b\).

If equation \(2x^3 + ax^2 + bx + 4 = 0\) has \(3\) real roots, where \(a, b > 0,\) show that \(a + b > -6\).

Show that equation \(x^3 + 2x^2 + x + 5 = 0\) has only one real root \(\alpha\) such that \([\alpha] = -3,\) where \([x]\) denotes the integral part of \(x.\)

Solve \((x^2 + 2)^2 + 8x^2 = 6x(x^2 + 2)\)

Solve \(3x^3 = (x^2 + \sqrt{18}x + \sqrt{32})(x^2 - \sqrt{18}x - \sqrt{32}) - 4x^2\)

Solve \((15 + 4\sqrt{14})^t + (15 - 4\sqrt{14})^t = 30\) where \(t = x^2 - 2|x|\)

For \(a \le 0,\) determine all the roots of the equation \(x^2 - 2a|x - a| - 3a^2 = 0\)

Find all solutions of equation \(|x^2 - x - 6| = x + 2,\) where \(x\) is a real number.

Solve the equation \(2^{|x + 2|} - |2^{x + 1} - 1| = 2^{x + 1} + 1\)

Solve \(3^x + 4^x + 5^x = 6^x\)

Solve \((\sqrt{2 + \sqrt{3}})^x + (\sqrt{2 - \sqrt{3}})^x = 2^x\)

Let \(\{x\}\) and \([x]\) denote the fractional and integral part of a real number \(x\) respectively. Solve \(4\{x\} = x + [x]\)

For the same notation as previous problem solve \([x]^2 = x(x - [x])\)

Solve \(x^3 - y^3 = 127, x^2y - xy^2 = 42\)

Solve the system of equations: \(x - 2y + z = 0, 4x - y - 3z = 0, x^2 - 2xy + 3xz = 14\)

Solve \(x^4 + y^4 = 82, x+ y = 4\)

Solve \(\sqrt{a(2^x - 2) + 1} = 1 - 2^x, x \in R\)

If \(x\) is an integer, find the integral values of \(m\) satisfying the equation \((x - 5)(x + m) + 2 = 0\)

Find the positive solutions of the system of equations \(x^{x + y} = y^n\) and \(y^{x + y}= x^{2n}y^n\) where \(n > 0\)

Solve the equation \((144)^{|x|} - 2(12)^{|x|} + a = 0\) for every value of the parameter \(a.\)

If \(m\) and \(n\) are odd integers show that the equation \(x^2 + 2mx + 2n = 0\) cannot have rational roots.

If \(f(x) = ax^3 + bx^2 + cx + d\) has local extrema at two points of opposite sign, then prove that the roots of the quadratic equation \(ax^2 + bx + c = 0\) are real and distinct.

If \(a\) and \(b\) are real, \(b\ne 0,\) prove that the roots of the quadratic equation \(\frac{(x - a)(ax - 1)}{x^2 - 1} = b,\) can never be equal.

If \(n\) and \(r\) are positive integers such that \(0 < r < n,\) then show that the roots of the quadratic equation \(^nC_rx^2 + 2~^nC_{r + 1}x + ^nC_{r + 2} = 0\) are real.