Quadratic Equation

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:

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

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…

Let$a,b,c$be the length of three sides of a triangle satisfying the condition$\left(a^{2}+b^{2}\right)x^{2}-2b\left(a+c\right)$$x+\left(b^{2}+c^{2}\right)=0$. If the set of all possible values of$x$ is in the…

Let $p,q\in\,Q.$ If $2-\sqrt{3}$ is a root of the quadratic equation $x^{2}+px+q=0,$ then

If $\alpha\,,\beta\,$ and $\gamma\,$ are three consecutive terms of a non-constant G.P. Such that the equations $\alpha\,x^{2}+2\beta\,x+\gamma\,=0$ and $x^{2}+x-1=0$ have a common root, then…

The value of $\lambda\,$ such that sum of the squares of the roots of the quadratic equation, $x^{2}+\left(3-\lambda\,\right)x+2=\lambda\,$ has the least value is:

The equation $\sqrt{3x^{2}+x+5}=x-3$, where $x$is real, has

If $\lambda\,$ be the ratio of the roots of the quadratic equation in $x,3m^{2}x^{2}+m\left(m-4\right)x+2=0$, then the least value of $m$ for which $\lambda\,+\frac{1}{\lambda\,}=1$, is :

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_____?

If the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a}) \mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and…

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