What is the probability that both roots of the equation $𝐴𝑥^2+𝐵𝑥+𝐶=0$ are real?

enter image description here> Let 𝐴, 𝐵, and 𝐶 be independent random variables, uniformly

distributed over $[0,5], [0,1]$ and $[0,2]$ respectively.

What is the probability that both roots of the equation $𝐴𝑥^2+𝐵𝑥+𝐶=0$ are real?

I know I have to do triple integral but I have zero clues how to start.

Following Polya's advice let's break this down into steps and then solve each step:

• How do we know if the roots of $ax^2 + bx + c=0$ are real ? The roots are real if the discriminant $b^2-4ac$ is greater than or equal to $0$.
• We are told $a,b,c$ are uniformly distributed within the cuboid region $0 \le a \le 5; 0\le b \le 1; 0 \le c \le 2$. So we can re-state the problem as "in what proportion of the cuboid is $b^2-4ac \ge 0$ ?".
• We know that the volume of the cuboid is $10$. So we need to find the volume of the cuboid in which $b^2-4ac \ge0$ and then divide this by $10$.
• Let's consider a simpler two-dimensional problem. Suppose $c$ is fixed at some value $C$ (so we are taking a slice of the cuboid). In what area within the rectangle $0 \le a \le 5; 0\le b \le 1$ is $b^2-4aC \ge 0$ ?
• We can re-state this again. For a fixed value $C$, in what area within the rectangle $0 \le a \le 5; 0\le b \le 1$ is $\displaystyle a \le \frac {b^2} {4C}$ ?

• Sketch a diagram. From the sketch it becomes clear that we need to find the area $A$ between the curve $\displaystyle a = \max (\frac {b^2} {4C}, 5)$ and the line $a=0$ that is also between the lines $b=0$ and $b=1$. We can do find by integration:

  $\displaystyle A(C) = \int_0^1 \max (\frac {b^2} {4C}, 5) db$

• This gives us the area $A(C)$ for a slice $c=C$ of the cuboid. To find the volume across the whole cuboid we have to integrate $A(c)$ between $c=0$ and $c=2$. Then divide by 10 to find the answer to the original problem.

You start with this: given a quadratic equation, how can you tell whether both roots are real? There is (a part of) a very well-known formula you can use.

Once you have that, you see that this condition will describe part of the box defined by the domains of $A,B,C$. Find the volume of that part, and you're very nearly done.