The given image shows the graph of a quadratic equation $y = ax^2 + bx + c$. Based on the graph,which of the following is true?

If one root of the equation $x^2 + px + 12 = 0$ is $4$,and the equation $x^2 + px + q = 0$ has equal roots,then what is the value of $q$?

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

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$ where $ac \neq 0$,then $P(x) \cdot Q(x) = 0$ has at least:

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