Animated Solution for Mathematics - Quadratic Equations: Product of real roots of the equation $x^2+|x|+9=0$

My dear students, today we are going to dismantle a classic trap. At first glance, the equation $x^2+|x|+9=0$ looks like a standard quadratic equation that might be begging for the quadratic formula or Vieta's relations.

But in the world of JEE Advanced, the first step is always to pause and observe the soul of the equation. Let us peel back the layers of this problem together.

Look closely at the expression $x^2+|x|+9$. We have three distinct components.

First, consider $x^2$. For any real number $x$, the square is always non-negative. Whether $x$ is a large positive number or a large negative number, $x^2$ will always be $\ge 0$.

Next, look at the absolute value term, $|x|$. By definition, the modulus function represents the distance from the origin, which can never be negative. Thus, $|x|\ge 0$.

When we combine these two, we are adding two non-negative quantities. Therefore, $x^2+|x|\ge 0$.

Now, let us introduce the constant term, $9$. If we add $9$ to our inequality, we get:

This is the moment of clarity. We have just proven that the expression $x^2+|x|+9$ can never be less than $9$.

Imagine a graph of the function $y=x^2+|x|+9$. The lowest point of this curve, the vertex, sits at $y=9$ when $x=0$.

As $x$ moves away from zero, the value of $y$ only increases. The curve is essentially a U-shaped structure that floats entirely above the line $y=9$.

A real root of an equation is defined as an $x$-value where the function equals zero, or graphically, where the curve intersects the $x$-axis ($y=0$).

But look at our function: it is trapped in the region where $y\ge 9$. It never descends to touch the $x$-axis.

There is an unbridgeable gap of $9$ units between our function and the $x$-axis. Because the curve never intersects the $x$-axis, the equation $x^2+|x|+9=0$ has no real roots.

Consequently, when the question asks for the product of the real roots, we must realize that the set of real roots is empty. A product of an empty set of numbers does not exist.

Do not let the presence of an equation tempt you into blind calculation. Logic is your most powerful tool. The answer is that the product does not exist.