Animated Solution for Mathematics - Basic Mathematics: The number of real solutions of the equation $x(x^2+3|x|+5|x-1|+6|x-2|)=0$ is

Seeing Through the Complexity

Imagine you are standing before a massive, intimidating mountain. If you look at the whole thing, you might feel overwhelmed. But if you look at the path, step by step, you realize it is just a series of small, manageable climbs.

This equation, $x(x^2+3|x|+5|x-1|+6|x-2|)=0$, is exactly like that mountain. It looks like a tangled mess of absolute values, but the moment you apply the right perspective, the path becomes clear.

The first thing an elite student does is pause. Do not rush to expand the brackets! That is the trap.

Instead, look at the structure. We have a product of two factors: $A=x$ and $B=(x^2+3|x|+5|x-1|+6|x-2|)$.

The Zero Product Property tells us that $A\cdot B=0$ if and only if $A=0$ or $B=0$. This is our 'divide and conquer' strategy. We have split one terrifying problem into two simple ones.

The first case is trivial: $x=0$. That is our first solution. We have one in the bag!

The Second Factor

Now, we turn our attention to the second factor:

We need to know if this expression can ever be zero. Let's look at the soul of each term.

First, $x^2$. For any real number $x$, $x^2$ is always non-negative. If $x eq 0$, it is strictly positive.

Next, look at the absolute value terms: $3|x|$, $5|x-1|$, and $6|x-2|$. By definition, the absolute value of any real number is always greater than or equal to zero. It represents a distance, and distance cannot be negative.

So, we are looking at a sum of terms where one is strictly positive (for $x eq 0$) and the others are non-negative.

Why It Never Touches Zero

Think about what happens when you add these up. If you have a strictly positive number and you add non-negative numbers to it, the result must be strictly positive. It can never be zero.

What about the case where $x=0$? We already checked that. If $x=0$:

Since $17 eq 0$, the second factor is never zero. The graph of $f(x)$ stays strictly above the x-axis. It never touches it.

Therefore, the only real solution is the one we found at the very beginning: $x=0$. We have conquered the mountain. The total number of real solutions is exactly 1.

Remember, in JEE Advanced, the most elegant solution is often the one that relies on the fundamental properties of numbers rather than brute-force calculation. Keep this clarity of mind, and you will solve any problem that comes your way.