If $8$ and $2$ are the roots of ${x^2} + ax + \beta = 0$ and $3$ and $3$ are the roots of ${x^2} + \alpha x + b = 0$,then the roots of ${x^2} + ax + b = 0$ are

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-x-1=0$. If $p_{k}=(\alpha)^{k}+(\beta)^{k}, k \geq 1,$ then which one of the following statements is not true?

If $\alpha, \beta$ are the roots of $(x - a)(x - b) = c$,where $c \neq 0$,then the roots of $(x - \alpha)(x - \beta) + c = 0$ are

Which of the following quadratic equations has real roots $x_1, x_2$ that satisfy the conditions $x_1^2+x_2^2=5$ and $3(x_1^5+x_2^5)=11(x_1^3+x_2^3)$?

Let $p(x)$ be a quadratic polynomial with constant term $1$. Suppose $p(x)$, when divided by $x-1$, leaves remainder $2$ and when divided by $x+1$, leaves remainder $4$. Then, the sum of the roots of $p(x)=0$ is

If $\alpha, \beta, \gamma$ are the roots of $x^3+ax^2+bx+c=0$,then find the value of $\sum \frac{1}{\alpha}$,given that $\alpha, \beta, \gamma$ are non-zero.