Animated Solution for Mathematics - Quadratic Equations: The equation $\frac{x}{x-1}+\frac{2}{x-1}=1-\frac{2}{x-1}$ has

Welcome, fellow traveler of the JEE landscape. Today, we encounter a problem that seems deceptively simple.

You look at the equation

and your brain immediately wants to start cross-multiplying or clearing denominators. But wait—before we rush into the fray, let us pause. In the world of competitive mathematics, the most dangerous traps are the ones that look like standard paths.

Before we touch a single variable, we must acknowledge the 'forbidden zone.' Look closely at the denominators; we see $(x-1)$ appearing repeatedly.

In the realm of real numbers, division by zero is the ultimate taboo. Therefore, we must declare our domain constraint immediately: $x eq 1$.

If we ever find a solution, it must not be $1$. If it is, we must discard it. This is the first step of a disciplined mathematician.

Now, let us organize our battlefield. On the Left Hand Side (LHS), we have

Since the denominators are already identical, we can combine them with elegance:

On the Right Hand Side (RHS), we have $1-\frac{2}{x-1}$. To make this comparable to our LHS, we express $1$ as $\frac{x-1}{x-1}$.

Now, the RHS becomes

We have now reduced the complex-looking equation to a simple comparison:

Since we have already established that $x eq 1$, we are mathematically permitted to multiply both sides by $(x-1)$. This leaves us with the numerators:

Take a deep breath. If we subtract $x$ from both sides, we are left with the stark, undeniable statement:

Is $2$ equal to $-3$? Of course not. This is a mathematical contradiction.

In the language of logic, this tells us that our initial premise—that there exists some $x$ such that the LHS equals the RHS—is fundamentally flawed.

Imagine these two functions as two parallel paths on a map. No matter how far you travel along the $x$-axis, these two curves will never meet.

The Beauty of 'No Root'

Many students feel a sense of panic when they don't find a numerical answer like $x=5$ or $x=-2$. They think, 'Did I do something wrong?'

But in JEE Advanced, finding 'No Root' is just as valid and just as powerful as finding a specific value. You have successfully navigated the trap, respected the domain, and followed the logic to its inevitable conclusion.

You haven't failed to find a root; you have proven that none exists. That, my friend, is the true essence of mathematical mastery.