Assume that we have the following polynomial:
$$ax^2 + bx =c$$
and a, b, c are i.i.d uniform random variables in [0, 1].
I'm trying to calculate the probability that the root is real, and that would be same as calculating the probability that
$$b^2 - 4ac \geq 0$$
which turns out to be slightly greater than $\frac{1}{4}$.
I was wondering... is there a formula that says if $a, b, c$ are i.i.d uniform random variables in [0, 1], and k is a positive real number greater than 1, then
$$P(b^2 - kac \geq 0) - \frac{1}{k} \leq C_k$$
where $C_k$ is a positive real number that's only dependent on $k$.
For $0 \le t \le 1$, $P(ac <= t) = 1 - \int_{t}^1 (1-t/s)\; ds = t - t \ln t$, so $ac$ has density $f_{ac}(t) = - \ln t$, $0 < t < 1$.
$P(b^2 - k a c \ge 0|ac) = 0$ if $ac > 1/k$, $1 - \sqrt{kac}$ otherwise. So $$P(b^2 - k a c \ge 0) = \int_0^{1/k} f_{ac}(t) (1 - \sqrt{kt})\; dt = \dfrac{\ln k}{3k} + \dfrac{5}{9k}$$