If $\alpha \neq 0$ and $0$ are the roots of the equation $x^2 - 5kx + (6k^2 - 2k) = 0$,then $\alpha = $

Let $a \ne a_1 \ne 0,$ $f(x) = ax^2 + bx + c,$ $g(x) = a_1x^2 + b_1x + c_1,$ and $p(x) = f(x) - g(x).$ If $p(x) = 0$ only for $x = -1$ and $p(-2) = 2,$ then the value of $p(2)$ is

The product of all the real roots of $x^2-8x+9-\frac{8}{x}+\frac{1}{x^2}=0$ is

The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?
$I$. It has at least one negative root.
$II$. It has at least one positive root.
$III$. Both its roots are real.

If the expression $\left( mx - 1 + \frac{1}{x} \right)$ is always non-negative for $x > 0$,then the minimum value of $m$ must be