Quadratic equations practice · Mixed difficulty

If $\alpha ,\beta$ are the roots of the equation ${x^2} - \left( {5 + {3^{\sqrt {\log _3^5} }} - {5^{\sqrt {\log _5^3} }}} \right)x + 3\left( {{3^{{{\left( {{{\log }_3}5} \right)}^{\frac{1}{3}}}}} - {5^{{{\left( {\log _5^3} \right)}^{\frac{2}{3}}}}} - 1} \right) = 0$ then the equation, whose roots are $\alpha+\frac{1}{\beta}$ and $\beta+\frac{1}{\alpha}$

The solution set of the equation $pq{x^2} - {\left( {p + q} \right)^2}x + {\left( {p + q} \right)^2} = 0$ is

If $\alpha$ and $\beta$ are the roots of $a x^2+b x+c=0$, then those of $a x^2+2 b x+4 c=0$ are

Let $\alpha, \beta$ be the roots of the equation $x^2+2 x+5=0$, then the value of $\frac{4 \beta^3-2 \beta^2}{\alpha^3+\alpha}$ is equal to

Let $\alpha ,\beta$ be the roots of the equation ${x^2} - px + r = 0$ and $\frac{\alpha }{2},\;2\beta$ be the roots of the equation ${x^2} - qx + r = 0$. Then the value of $r$ is

If $a{x^2} + bx + c = 0$ and $2{x^2} + 3x + 4 = 0$ have a common root where $a,\;b,\;c \in N$ (set of natural numbers), the least value of $a + b + c$ is

The number of real roots of the equation $\left( {\sin {2^x}} \right)\left( {\cos {2^x}} \right) = \frac{{{2^x} + {2^{ - x}}}}{2},$ is

If the quadratic equation $a x^2+b x+6=0$ does not have real roots and $b \in R^{+}$, then

The number of integer values of $x$ satisfying $(x+1)^2>5 x-1 \text { and }(x+1)^2<7 x-3$ simultaneously is

If $\alpha ,\;\beta$ are the roots of ${x^2} - ax + b = 0\;\,\mathrm{and\;if}\,\;{\alpha ^n} + {\beta ^n} = {V_n},$ then

The product of all the values of $x$ satisfying the equation $(5+2 \sqrt{6})^{x^2-3}+(5-2 \sqrt{6})^{x^2-3}=10$ is

Both the roots of the given equation $\left( {x - a} \right)\left( {x - b} \right) + \left( {x - b} \right)\left( {x - c} \right) + \left( {x - c} \right)\left( {x - a} \right) = 0$ are always

The values of &#39; $a$ &#39; for which one root of the equation $x^{2}-(a+1) x+a^{2}+a-8=0$ exceeds 2 and the other is lesser than 2 , are given by :

The solution set of the inequation $\left| {\frac{{2x - 1}}{{x - 1}}} \right| > 2,$ is

Given that $\tan A$ and $\tan B$ are the roots of ${x^2} - px + q = 0$, then the value of ${\sin ^2}\left( {A + B} \right)$ is

If $\alpha$ and $\beta$ are the roots of the equation $a{x^2} + bx + c = 0$ then, $\left( {1 + \alpha + {\alpha ^2}} \right)\;\left( {1 + \beta + {\beta ^2}} \right)$ is equal to

${\left( {{{\log }_5}x} \right)^2} + \left( {{{\log }_5}x} \right) < 2,$ then $x$ belongs to the interval

If $\alpha$ and $\beta$ are the roots of the equation $a x^2+b x+c=0$, then the equation whose roots are $\frac{1}{\alpha+\beta}, \frac{1}{\alpha}+\frac{1}{\beta}$ is

If $a,\;b$ and $c$ are in geometric progression and the roots of the equation $a{x^2} + 2bx + c = 0$ are $\alpha$ and $\beta$ and those of $c{x^2} + 2bx + a = 0$ are $\gamma$ and $\delta$, then

The sum of all real roots of the equation $|x-2|^2+|x-2|-2=0$ is

The value of $a$ for which the sum of the squares of the roots of the equation ${x^2} - \left( {a - 2} \right)x - a - 1 = 0$ assumes the least value, is

If ${e^{\cos x}} - {e^{ - \cos x}} = 4,$ then the value of $\cos x$ is

If $a_i > 0$ for $i=1,2, \ldots, n$ and $a_1 a_2 \ldots a_n=1$, then minimum value of $\left(1+a_1\right)\left(1+a_2\right) \ldots\left(1+a_n\right)$ is

If $\alpha, \beta$ are roots of $9 x^2-11 x+1=0$ then value of $\frac{1}{(9 \alpha-11)^2}+\left(\frac{11 \beta-1}{9}\right)$ is-

If $a$ and $b$ are the non-zero distinct roots of ${x^2} + ax + b = 0,$ then the least value of ${x^2} + ax + b$ is

If $\alpha + \beta = 4$ and ${\alpha ^3} + {\beta ^3} = 44$, then $\alpha ,\;\beta$ are the roots of the equation

Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots $(4,3)$. Rahul made a mistake in writing down coefficient of $x$ to get roots $(3,2)$. The correct roots of equation are

If $a,c \ne 0$ and $\alpha ,\beta$ are the roots of the equation $a{x^2} + bx + c = 0$, then the quadratic equation with $1/\alpha$ and $1/\beta$ as its root is

$\frac{x+4}{x-3} < 2$ is satisfied when $x$ satisfies

If one root of the quadratic equation $a{x^2} + bx + c = 0$ is equal to nth power of the other root, then the value of ${\left( {a{c^n}} \right)^{1/\left( {n + 1} \right)}} + {\left( {{a^n}c} \right)^{1/\left( {n + 1} \right)}}$ is equal to