(I've seen the other questions in this site similar to my problem, but they didn't help much). So, the problem is:
The numbers $B$ and $C$ are chosen at random between $−1$ and $1$,
independently of each other. What is the probability that the
quadratic equation $x^2+Bx+C=0$ has real roots?
I've actually gotten an answer, but it's wrong according to my book. Here's what I did: I know that the condition for that equation to have real roots is that $B^2-4C\ge 0\Rightarrow C\le\frac{B^2}{4}$. The sample space consists of all points $(x,y):-1\le x\le1,-1\le y\le1$. So I think the desired probability should be the ratio of the area a square of side-lenght $2$ to the area under $C\le\frac{B^2}{4}$ from $-1$ to $1$.
That makes intuitive sense to me, but the book says the answer is actually $\frac{1}{4}(2+\int_{-1}^{1}x^2dx)$. Where does that $2$ come from? Thanks very much in advance.
Best Answer
Note that the discriminant is always non negative if $C\leq 0$. You forgot the area under the $x$- respectively $B$-axis.