Quadratic Equations

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

Let $a,b\in\,R$be such that the equation $ax^{2}-2bx+15=0$has repeated root $\alpha\,$and if $\alpha\,$ and $\beta\,$ are the roots of the equation $x^{2}-2bx+21=0$,then $\alpha\,^{2}+\beta\,^{2}$ is equal to:

Consider the quadratic equation $\left(c-5\right)x^{2}-2cx+\left(c-4\right)=0,c\neq\,5.$ Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $\left(0,2\right)$ and…

The sum of all integral values of $k(k\neq\,0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is_____?

The equation $e^{4x}+8e^{3x}+13e^{2x}-8e^{x}+1=0,x\in\,R$ has :

Let $a,b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x^{2}-8ax+2a=0$and $q$ and $s$ are the roots of the equation $x^{2}+12bx+6b=0$, such that…

Let $\alpha\,$ and $\beta\,$ be the roots of the equation $x^{2}-x-1=0$ . If $p_{k}=\left(\alpha\,\right)^{k}+\left(\beta\,\right)^{k},k\geq\,1,$ then which one of the following statements is not true?

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

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…

Let$\left[t\right]$ denote the greatest integer$\leq\,t$. Then the equation in$x,\left[x\right]^{2}+2\left[x+2\right]-7=0$ has :

The sum of the solutions of the equation $\left|\sqrt{x}-2\right|+\sqrt{x}\left(\sqrt{x}-4\right)+2=0,\left(x>0\right)$ is equal to

If $\alpha\,,\beta\,\in\,R$ are such that $1-2i$(here $i^{2}=-1$) is a root of $z^{2}+\alpha\,z+\beta\,=0$, then $\left(\alpha\,-\beta\,\right)$ is equal to: