If $r$ and $s$ are positive,what is the nature of the roots of the quadratic equation $ax^2 - rx - s = 0$?

Suppose the height of a pyramid with a square base is decreased by $p \%$ and the lengths of the sides of its square base are increased by $p \%$ (where $p > 0$). If the volume remains the same,then:

Find the remainder when $x^4-11x^3+44x^2-76x+48$ is divided by $x^2-7x+12$.

If $x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots \infty}}}$,then $x =$

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

The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is