Relationship between Zeros 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

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, $\frac{{\alpha }^{34}{\beta }^{24}+{\alpha }^{32}{\beta }^{26}+2{\alpha }^{33}{\beta }^{25}}{{\alpha }^{31}{\beta }^{20}+{\alpha }^{28}{\beta }^{23}+3{\alpha }^{30}{\beta }^{21}+3{\alpha }^{29}{\beta }^{22}}=$

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

If the roots of the equation $x^2+bx+c=0$ are $\alpha \& \frac{1}{\alpha }$, then the value of $c$ is

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 $\tan 15°$ and $\tan 30°$ are the roots of the equation $x^2+px+q=0$, then $pq=$

Let $\alpha$ and $\beta$ are roots of equation $7x^2-5x-1=0$, then $\underset{n\to \infty }{\lim } \sum _{r=0}^n\left(\frac{1}{{\left(7\alpha -5\right)}^r}+\frac{1}{{\left(7\beta -5\right)}^r}\right)$ is:

Let $\alpha$ and $\beta$ be the roots of $6x^2-2x+1=0$ and $S_n={\alpha }^n+{\beta }^n$, then $\underset{n\to \infty }{\lim }\sum _{k=1}^n{S}_k$ 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 $\alpha$ and $\beta$ are the roots of the equation $2x^2+5x+k=0$, and $4\left({\alpha }^2+{\beta }^2+\alpha \beta \right)=23$, then which of the following is true?

If $\alpha$ and $\beta$ are the roots of equation $x^2-a(x+1)-b=0$, then $\frac{{\alpha }^2+2\alpha +1}{{\alpha }^2+2\alpha +b}+\frac{{\beta }^2+2\beta +1}{{\beta }^2+2\beta +b}=$