Let one root of $ax^2 + bx + c = 0$,where $a, b, c$ are integers,be $3 + \sqrt{5}$. Then the other root is:

If $(x-2)$ is a common factor of the expressions $x^2+ax+b$ and $x^2+cx+d$,then $\frac{b-d}{c-a}$ is equal to

If $x = 2 + \sqrt{3}$ and $xy = 1$,then $\frac{x}{\sqrt{2} + \sqrt{x}} + \frac{y}{\sqrt{2} - \sqrt{y}} = $

If $1-i$ is a root of the equation $x^2+ax+b=0$,then $b$ is equal to

Let $a, b \in \mathbb{R}, a \neq 0$ be such that the equation $a x^{2}-2 b x+5=0$ has a repeated root $\alpha,$ which is also a root of the equation $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation,then $\alpha^{2}+\beta^{2}$ is equal to:

Let $[x]$ be the greatest integer less than or equal to $x$ for a real number $x$. Then the equation $[x^2] = x + 1$ has: