Quadratic Equation

Let $S$, be the set of all real roots of the equation, $3^{x}\left(3^{x}-1\right)+2=\left|3^{x}-1\right|+\left|3^{x}-2\right|$, then

If for some $p,q,r\in\,R$, all have positive sign, one of the roots of the equation $\left(p^{2}+q^{2}\right)x^{2}-2q\left(p+r\right)x+q^{2}+r^{2}=0$is also a root of the equation $x^{2}+2x-8=0$, then…

Let $\alpha\,\text{and}\beta\,$ be the roots of equation $px^{2}+qx+r=0$, $p\neq\,0$. If $p,q,r$are in A.P.and $\frac{1}{\alpha\,}+\frac{1}{\beta\,}=4$, then the value of $\left|\alpha\,-\beta\,\right|$ is

The least positive value of ' $a$ ' for which the equation, $2x^{2}+\left(a-10\right)x+\frac{33}{2}=2a$ has real roots is ___________?

Let $\alpha\,$ and $\beta\,$ be two real numbers such that $\alpha\,+\beta\,=1$ and $\alpha\,\beta\,=-1$. Let $p_{n}=\left(\alpha\,\right)^{n}+\left(\beta\,\right)^{n}$,$p_{n-1}=11$and $p_{n+1}=29$ for some integer…

For a natural number $n$, let $\alpha\,_{n}=19^{n}-12^{n}$. Then, the value of $\frac{31\alpha\,_{9}-\alpha\,_{10}}{57\alpha\,_{8}}$is ______

Let$p,q$ and$r$be real numbers $\left(p\neq\,q,r\neq\,0\right)$, such that the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of…

If$\frac{1}{\sqrt{\alpha\,}},\frac{1}{\sqrt{\beta\,}}$are the roots of the equation$ax^{2}+bx+1=0,\left(a\neq\,0,a,b\in\,R\right)$,then the equation$x\left(x+b^{3}\right)+\left(a^{3}-3abx\right)=0$ has roots:

If three distinct numbers $a,b,c$ are in G.P. and the equations $ax^{2}+2bx+c=0$ and $dx^{2}+2ex+f=0$ have a common root, then which one of the following statements is correct?

The number of integral values of$m$for which the quadratic expression$\left(1+2m\right)x^{2}-2\left(1+3m\right)x+4\left(1+m\right),x\in\,R$is always positive, is

The real number $k$ for which the equation, $2x^{3}+3x+k=0$has two distinct real roots in $[0,1]$belongs to

Let $p$ and $q$ be two positive numbers such that $p+q=2$and $p^{4}+q^{4}=272$.Then $p$ and $q$ are roots of the equation: