Relations Between Roots and Coefficients

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

$\alpha ,\beta$ 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 of the roots of the equation $x^2+px+q=0$ is equal to the square of the other then

Suppose that the equation $a{x}^2+bx+c=0$ has roots $\alpha$ and $\beta$, both of which are different from $\frac{1}{3},$ then an equation whose roots are $\frac{1}{3\alpha -1}$ and $\frac{1}{3\beta -1}$ is

Let $a, b, c$ be positive real numbers. If $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ has two roots which are numerically equal but opposite in sign, then the value of $\text{'}m\text{'}$ is

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 $\alpha ,\beta$ 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

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

The equation formed by decreasing each root of $a{x}^2+bx+c=0$ by $1$ is $2x^2+8x+2=0$, then 

If one root of the equation $i{x}^2-2\left(1+i\right)x+\left(2-i\right)=0$ is $2-i$ , then the other root is

If $m$ is chosen in the quadratic equation $\left(m^2+1\right)x^2-3x+{\left(m^2+1\right)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

If one of the roots of$x^2-8x+k=0$ is $2$, then the other root is