The polynomial equation of degree $5$ whose roots are the translates of the roots of $x^5-2x^4+3x^3-4x^2+5x-6=0$ by $-2$ is:

The sum of the squares of all the roots of the equation $x^2+|2x-3|-4=0$ is:

For $0 < c < b < a$,let $(a+b-2c)x^2 + (b+c-2a)x + (c+a-2b) = 0$ and $\alpha \neq 1$ be one of its roots. Then,among the two statements:
$(I)$ If $\alpha \in (-1, 0)$,then $b$ cannot be the geometric mean of $a$ and $c$.
$(II)$ If $\alpha \in (0, 1)$,then $b$ may be the geometric mean of $a$ and $c$.

Let $\alpha, \beta$ be the roots of the equation $x^2 - |a|x - |b| = 0$ such that $|\alpha| < |\beta|$. If $|a| < \beta - 1$,then the positive root of $\log_{|\alpha|} \left( \frac{x^2}{\beta^2} \right) - 1 = 0$ is

Evaluate $\sqrt{x + 2\sqrt{x - 1}} + \sqrt{x - 2\sqrt{x - 1}}$ for $x \ge 1$.

If the equation whose roots are $p$ times the roots of the equation $x^4-2ax^3+4bx^2+8ax+16=0$ is a reciprocal equation,then $|p|=$ :