[Math] Find the probability that the quadratic equation $x^2+Bx+C=0$ has real roots.

Question

Let $B$ be the coefficient of $x$, and $C$ be the constant term in the quadratic equation:$x^2+Bx+C=0.$ Assume that $B$ is a uniform random variable on the interval $\left(1,3\right)$, $C$ is a uniform random variable on the interval $\left(1,3\right)$, and that $B$ and $C$ are independent. Thus, the joint PDF of the random pair $\left(B,C\right)$ is $f(b,c)=f_B(b)f_Cf(c)=(1/2)(1/2)=(1/4)$
when $(b,c)\in (1,3)\times(1,3)$; and $0$ otherwise. Find the probability that the quadratic equation $x^2+Bx+C=0$ has real roots.

Answer 1

Hint: the condition for the quadratic equation to have real roots is $B^2 - 4 C \ge 0$. So what is the probability of that?