Quadratic Equation and its Solutions

Which of the following equations has real roots$?$

The roots of equation $y^3+y^2-y-1=0$ are

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 $:$

Match the List $\mathrm{I}$ and List $\mathrm{II}$ and select the correct answer using code:

Match the List $\mathrm{I}$ and List $\mathrm{II}$ and select the correct answer using code:

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