Relation Between Roots and Coefficients of the Quadratic Equation

If $\alpha , \beta$ are the nonzero roots of $a{x}^2+bx+c=0$ and $\alpha +h, \beta +h$ are the roots of $p{x}^2+qx+r=0$, then $h$

A value of $b$ for which the equations $x^2+bx-1=0$ and $x^2+x+b=0$ have one root in common is?

If $\alpha$ and $\beta$ are the roots of the equation $x^2-6x+12=0$and the value of ${\left(\alpha -2\right)}^{24}-\frac{{\left(\beta -6\right)}^8}{{\alpha }^8}+1$ is $4^a$, then the value of $a$ is $\_\_\_\_\_\_\_\_\_$

Suppose $a, b, c$ are the roots of the cubic $x^3-x^2-2=0$. Then the value of $a^3+b^3+c^3$ is                 .

The quadratic equation $P\left(x\right)=0$ with real coefficients has purely imaginary roots. Then the equation $P\left(P\left(x\right)\right)=0$ has 

If $a{x}^2+bx+c=0$ has imaginary roots and $a-b+c>0,$ then the set of points $\left(x,y\right)$ satisfying the equation $\left|a\left(x^2+\frac{y}{a}\right)+\left(b+1\right)x+c\right|=\left|a{x}^2+bx+c\right|+\left|x+y\right|$ consists of the region in the $xy$-plane which is

The set of values of $a$ for which $\left(a-1\right)x^2-\left(a+1\right)x+a-1\ge 0$ is true for all $x\ge 2$ is

For the quadratic equation $x^2+2\left(a+1\right)x+9a-5=0$, which of the following is/are true?

If $\frac{x^2+5}{2}=x-2\cos \left(m+nx\right)$ has at least one real root, then

If the equation $x^2-3px+2q=0$ and $x^2-3ax+2b=0$  have a common root and the other root of the second equation is the reciprocal of the other root of the first, then ${\left(2q-2b\right)}^2$ is equal to 

If $\alpha , \beta$ are the roots of $x^2+px+q=0$ and $\gamma , \delta$ are the roots of $x^2+px+r=0$, then $\frac{\left(\alpha -\gamma \right) \left(\alpha -\delta \right)}{\left(\beta -\gamma \right) \left(\beta -\delta \right)}=$

If $\alpha \mathrm{and} \beta$ be the roots of the equation $x^2+px-\frac{1}{\left(2p^2\right)}=0$. Where $p \in R$. Then the minimum value of ${\alpha }^4+{\beta }^4$ is

If one root of $x^2-x-k=0$ is square of the other, then the value of $k$ is 

If the roots of the equation $a{x}^2+bx+c=0$ are of the form $\frac{\left(k+1\right)}{k} \mathrm{and} \frac{\left(k+2\right)}{\left(k+1\right)} \mathrm{then} {\left(a+b+c\right)}^2$ is equal to

If $\alpha , \beta$ are the nonzero roots of $a{x}^2+bx+c=0$ and ${\alpha }^2, {\beta }^2$ are the roots of $a^2{x}^2+b^{2}x+c^2=0$, then $a, b, c$ are in

If $a{\left(p+q\right)}^2+2bpq+c=0$ and $a{\left(p+r\right)}^2+2bpr+c=0; \left(a\ne 0\right)$, then

If $\alpha , \beta$ are the roots of $a{x}^2+c=bx$, then the equation ${\left(a+cy\right)}^2=b^{2}y \mathrm{in} y$ has the roots

The quadratic $x^2+ax+b+1=0$ has roots which are positive integers, then $\left(a^2+b^2\right)$ can be equal to

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

If $\alpha , \beta$ are the roots of $x^2-px+q=0$ and ${\alpha }^{\text{'}}, {\beta }^{\text{'}}$are the roots of $x^2-p^{\text{'}}x+q^{\text{'}}=0$, then the value of ${\left(\alpha -{\alpha }^{\text{'}}\right)}^2+{\left(\beta -{\alpha }^{\text{'}}\right)}^2+{\left(\alpha -{\beta }^{\text{'}}\right)}^2+{\left(\beta -{\beta }^{\text{'}}\right)}^2$ is