Relation 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

Let $x^2+6\sqrt{x}+4=0$ be any quadratic equation and $\alpha ,\beta$ are roots of that equation then, $\frac{{\alpha }^{34}{\beta }^{24}+{\alpha }^{32}{\beta }^{26}+2{\alpha }^{33}{\beta }^{25}}{{\alpha }^{31}{\beta }^{20}+{\alpha }^{28}{\beta }^{23}+3{\alpha }^{30}{\beta }^{21}+3{\alpha }^{29}{\beta }^{22}}=$

If roots of $x^2+{\left(60\right)}^{\frac{1}{4}}x+a=0$ be $\alpha \& \beta$ and ${\alpha }^4+{\beta }^4=30$, then find the product of all possible values of $a$.

The roots of the equation $x^2+px+q=0$ are $p$ and $q$ such that $p\ne 1$, then

If $a,\beta$ are roots of the equation $4x^2+2x-1=0$ then ${\alpha }^4+{\beta }^4+4{\alpha }^2+2\alpha$ is equal to

If $a{x}^2-2bx+15=0, a,b\in R$ had repeated roots $\alpha$ and the equation $x^2-2bx+21=0$ had roots $\alpha$ and $\beta$, then ${\alpha }^2+{\beta }^2$ is

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ then ${\left(\frac{\alpha }{a\beta +b}\right)}^3-{\left(\frac{\beta }{a\alpha +b}\right)}^3=$

Let $\alpha ,\beta$ are real number such that ${\alpha }^2,\beta$ are the roots of the equation $x^2-px+8=0$ and $\alpha ,{\beta }^2$ are the roots of the equation $x^2-qx+1=0$

then $p+q$ is

If one of the root of the equation $x^2+xf\left(a\right)+a=0$ is the cube of the other for all $x\in R$ then $f\left(x\right)$

If the difference of the squares of the roots of the equation $x^2-6x+q=0$ is $24$, then the value of $q$ 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

Out of the following equations, find the equation having the sum of its roots $5$.

The coefficient of $x$ in the equation $x^2+ax+b=0$ was wrongly written as $17$ in place of $13$ and the roots thus found were $-2$ and $-15$. If $P$ and $Q$ $\left(Q>P\right)$ are the correct roots of the given quadratic equation, then the value of $\left|\frac{2P}{Q-1}\right|$ is equal to?