Formation of a Quadratic Equation

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

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

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

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 sum of roots of a quadratic equation is twice the product of the roots and the product of roots is an even prime number, then the quadratic equation obtained by doubling the roots of the earlier quadratic equation is given by

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 ${\left(\frac{\alpha }{a\beta +b}\right)}^3-{\left(\frac{\beta }{a\alpha +b}\right)}^3=$

The quadratic equations whose roots $x_1$, and $x_2$ satisfy the condition $x_1^2+x_2^2=5, 3\left(x_1^5+x_2^5\right)=11\left(x_1^3+x_2^3\right)$ (assume that $x_1, x_2$ are real) are

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