If $\alpha$ and $\beta$ are the roots of the equation $2x(2x+1)=1$,then $\beta$ is equal to

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$,with $\alpha>\beta$. For all positive integers $n$,define $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$ and $b_1=1$ and $b_n=a_{n-1}+a_{n+1}, n \geq 2$. Then which of the following options is/are correct?
$(1)$ $\sum_{i=1}^{n} a_i = a_{n+2}-1$ for all $n \geq 1$
$(2)$ $\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \frac{10}{89}$
$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n} = \frac{8}{89}$
$(4)$ $b_n = \alpha^n+\beta^n$ for all $n \geq 1$

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 + 6x + k = 0$,then the maximum value of $\left[\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right]$ when $k < 0$ is (where $[\cdot]$ denotes the greatest integer function)

If $1, 2, 3$ and $4$ are the roots of the equation $x^4+ax^3+bx^2+cx+d=0$,then $a+2b+c$ is equal to

If the difference between the roots of the equation $(k - 2)x^2 - (k - 4)x - 2 = 0$ is $3$, find the value of $k$.

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