Quadratic Equation and its Solutions

If   $\alpha , \beta$   are the roots of $x^2-px+1=0$, and $\gamma , \delta$ are roots of $x^2+qx + 1 = 0$,  then $\left(\alpha -\gamma \right)\left(\beta -\gamma \right)\left(\alpha +\delta \right)\left(\beta +\delta \right)$ is equal to

If $\alpha ,\beta$ are the roots of the equation, $\left(x-a\right)\left(x-b\right)+c=0$, find the roots of the equation, $\left(x-\alpha \right)\left(x-\beta \right)=c$

If one root of the equation $x^2+px+q=0$  is the square of the other, then

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 ${\left(a{c}^n\right)}^{\frac{1}{n+1}}+{\left(a^{n}c\right)}^{\frac{1}{n+1}}$ is equal to

A quadratic polynomial$y = f\left(x\right)$ satisfies $f\left(x\right)={\left[\frac{f\left(x+1\right)-f\left(x-1\right)}{2}\right]}^2$ for all real x. then the value of $\left[f\left(0\right)-f\left(-1\right)\right]+\left[f\left(0\right)-f\left(1\right)\right]$ is

$\alpha ,\beta$ are the roots of the equation $k\left(x^2-x\right)+x+5=\mathrm{0}$. If $k_1$ and $k_2$ are the two values of $k$ for which the roots $\alpha ,\beta$ are connected by the relation $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{4}{5}$. Find the value of $\frac{k_1}{k_2}+\frac{k_2}{k_1}$.

In a triangle  $PQR,\angle R=\frac{\pi }{2},$ if $\tan \left(\frac{P}{2}\right)$ and $\tan \left(\frac{Q}{2}\right)$  are distinct the roots of the equation  $a{x}^2+bx+c=0(a\ne 0),$ then –

In a triangle $PQR,\angle R=\frac{\pi }{2}, \text{if} \tan \left(\frac{P}{2}\right) \text{and} \tan \left(\frac{Q}{2}\right)$ are the roots of the equation  $a{x}^2+bx+c=0\left(a\ne 0\right)$ then

If one root of the equation  $x^2+px+q=0$ is square of the other root, then

$\sqrt{-2+2\sqrt{-2+2\sqrt{-2+\dots \dots \dots \infty }}}$ is equal to $:$

If $2\alpha$ is a root of $a{x}^2+bx+c=0, \beta$ is a root of $a{x}^2-2bx-c=0$, and the real numbers $\mathrm{a}, b, c\left(a>0\right)$, are such that $\beta <\alpha$, then a root $\gamma$ of $a{x}^2+4bx+2c=0$ always satisfies

If the product of the roots of the equation $x^2-5kx+2e^{2{\log }_e\left|k\right|}-1=0$ is $49,$ then the sum of the squares of the roots of the equation is

If $\alpha$ and $\beta$ are roots of equation $x^2-\mathrm{ }6x+a=0$ and satisfying relation $3\alpha +2\beta =16$ then the value of $\prime a\prime$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2-6x+a=0$ and satisfy the relation $3\alpha +2\beta =\mathrm{16 }$, then the value of $a$ is

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

The roots of the equation $a\left(b-2c\right)x^2+b\left(c-2a\right)x+c\left(a-2b\right)=0$ are, when $ab+bc+ca=0$

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ and $\alpha +h, \beta +h$ are the roots of $p{x}^2+qx+r=0$, then the value of $h$ is

Let $2{\sin }^{2}x+3\sin x-2>0$ and $x^2-x-2<0$, ($x$ is measured in radians). Then $x$ lies in the interval

If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $a{x}^2+bx+c=0$$,$ then