Animated Solution for Mathematics - Quadratic Equations: If $p$ and $q$ are chosen randomly from the set $\{1,2,3,4,5,6,7,8,9,10\}$, with replacement, determine the probability that the roots of the equation $x^2+px+q=0$ are real.

Welcome, future engineer. Today, we are not just solving a probability problem; we are exploring the architecture of quadratic equations.

When you see $x^2+px+q=0$, do not just see an equation. See a machine that generates roots.

Sometimes those roots are real, tangible numbers; sometimes they are imaginary, existing only in the complex plane. Our job is to find the probability that this machine produces real roots when we pick $p$ and $q$ from the set $\{1,2,\dots ,10\}$.

The Grid of Possibilities

Imagine a $10\times 10$ grid. The x-axis is $p$, and the y-axis is $q$. Every point $(p,q)$ on this grid is a potential universe.

Since we are choosing with replacement, we have $10$ choices for $p$ and $10$ choices for $q$. The total number of universes is $10\times 10=100$.

This is our denominator. It is the solid ground we stand on.

The Gatekeeper

What determines if the roots are real? It is the discriminant, $D=b^2-4ac$.

In our case, $a=1$, $b=p$, and $c=q$. So, $D=p^2-4q$.

For the roots to be real, we need $D\ge 0$, which leads us to the beautiful inequality:

This is our gatekeeper. Any point $(p,q)$ that satisfies this inequality allows the roots to be real. Any point that fails it sends the roots into the complex realm.

The Systematic Siege

We do not need to guess. We count. Let us walk through the values of $p$ systematically:

- For $p=1$: $1^2\ge 4q\Rightarrow 1\ge 4q\Rightarrow q\le 0.25$. Since $q\ge 1$, there are $0$ solutions.

- For $p=2$: $2^2\ge 4q\Rightarrow 4\ge 4q\Rightarrow q\le 1$. Only $q=1$ works. That is $1$ solution.

- For $p=3$: $3^2\ge 4q\Rightarrow 9\ge 4q\Rightarrow q\le 2.25$. So $q\in \{1,2\}$. That is $2$ solutions.

- For $p=4$: $4^2\ge 4q\Rightarrow 16\ge 4q\Rightarrow q\le 4$. So $q\in \{1,2,3,4\}$. That is $4$ solutions.

- For $p=5$: $5^2\ge 4q\Rightarrow 25\ge 4q\Rightarrow q\le 6.25$. So $q\in \{1,2,3,4,5,6\}$. That is $6$ solutions.

- For $p=6$: $6^2\ge 4q\Rightarrow 36\ge 4q\Rightarrow q\le 9$. So $q\in \{1,2,\dots ,9\}$. That is $9$ solutions.

The Shortcut

Now, look at $p=7$. Here, $p^2=49$.

Since the maximum value of $q$ is $10$, the maximum value of $4q$ is $40$. Since $49>40$, the inequality $p^2\ge 4q$ is satisfied for all $q$ from $1$ to $10$.

This holds for $p=7,8,9,10$. That is $4$ groups of $10$ solutions, giving us $40$ solutions instantly.

Adding them up: $0+1+2+4+6+9+40=62$.

Our probability is:

You have navigated the grid, respected the gatekeeper, and found the truth. This is the essence of JEE Advanced preparation: systematic, logical, and deeply rewarding.