Probability that the given quadratic equation has real solutions

Question

$a$ and $b$ are randomly chosen real numbers in the interval $[0, 1]$, that is both $a$ and $b$ are standard uniform random variables. Find the probability that the quadratic equation $x^2 + ax + b = 0 $ has real solutions?

Answer 1

We require the discriminant to be positive so $a^2-4b >0$. This corresponds to the region below the red curve. This area can easily be calculated using integration.

$\int_0^1 \frac{a^2}{4} da = \frac{1}{12}$