The equation x² - 4x + [x] + 3 = x[x], where [x] denotes the greatest integer function, has: (1) exa

Q: Why can't I just divide both sides by $(x-1)$? It makes the equation so much simpler!

Ah, the siren song of simplification! If you divide by $(x-1)$, you are implicitly assuming that $x \neq 1$. If $x=1$, you are dividing by zero, which is undefined. By dividing, you effectively delete the root $x=1$ from your solution set. Always factorize and set to zero instead of dividing by variables.

Q: I got $x - [x] = 3$. How does this relate to the fractional part?

Excellent observation! The expression $x - [x]$ is the fundamental definition of the fractional part of $x$, denoted as $\{x\}$. It represents the 'leftover' decimal part of a number after removing its integer component. So, your equation is simply asking: 'When is the fractional part of $x$ equal to 3?'

Q: Wait, why is $\{x\} = 3$ impossible? Can't the fractional part be any number?

Think about the definition. The fractional part $\{x\}$ is defined as $x - [x]$. Since $[x]$ is the largest integer less than or equal to $x$, the difference $x - [x]$ must always be in the interval $[0, 1)$. It can be $0$ (if $x$ is an integer), but it can never reach $1$, let alone $3$. It is physically impossible for the fractional part to exceed $1$.

Question

The equation $x^2 - 4x + [x] + 3 = x[x]$, where $[x]$ denotes the greatest integer function, has:

Options:

(A) exactly two solutions in $(-\infty, \infty)$

(B) no solution

(C) a unique solution in $(-\infty, 1)$

(D) a unique solution in $(-\infty, \infty)$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with solving the equation:
$$x^2 - 4x + [x] + 3 = x[x]$$

At first glance, this appears to be a complex mix of polynomials and step functions. However, in the context of JEE Advanced, we can resolve this by systematically separating the algebraic terms from the greatest integer function.

### The Strategy of Separation

To begin, we rearrange the equation to isolate the terms involving $[x]$ on one side:
$$x^2 - 4x + 3 = x[x] - [x]$$

By moving the $[x]$ term to the right, we create a structure that is ready for factorization. On the left, we have a standard quadratic expression, and on the right, we have a common factor of $[x]$.

### The Trap of Simplification

Factoring both sides, we obtain:
$$(x-1)(x-3) = [x](x-1)$$

A common mistake is to divide both sides by $(x-1)$. This is a mathematical error because it assumes $x 
eq 1$ and risks losing a valid solution. Instead, we bring all terms to one side:
$$(x-1)(x-3) - [x](x-1) = 0$$

Factoring out the common term $(x-1)$, we arrive at:
$$(x-1)(x - 3 - [x]) = 0$$

This product equals zero if and only if at least one of the factors is zero.

### The Fractional Part Revelation

**Case 1:** $x - 1 = 0$
This immediately yields the solution **$x = 1$**.

**Case 2:** $x - 3 - [x] = 0$
Rearranging this gives:
$$x - [x] = 3$$

Recall that the expression $x - [x]$ is the definition of the fractional part of $x$, denoted as $\{x\}$. By definition, the fractional part function is bounded such that $0 \le \{x\} < 1$.

Since the value $3$ falls outside the interval $[0, 1)$, it is impossible for $\{x\}$ to equal $3$. Therefore, Case 2 yields no real solutions.

### The Final Verdict

We have navigated the logic and avoided the common pitfalls of division. Case 1 provided the solution $x = 1$, while Case 2 provided no solutions.

Thus, the equation has a unique solution in the set of real numbers:
**$x = 1$**

Animated Solution for Mathematics - Functions: The equation $x^{2} - 4 x + [x] + 3 = x [x]$ , where $[x]$ denotes the greatest integer function, has:

Select Answer:

Visualized Solution

The Given Equation

Rearranging Terms

Factorizing the LHS

Factorizing the RHS

Equating the Factors

Factoring out $(x - 1)$

Case 1: $x - 1 = 0$

Case 2: $x - 3 - [x] = 0$

Fractional Part Function

Range of Fractional Part

Final Conclusion

The Sigma Insight: Classification of Functions

Analyzing the Setup

The Strategy of Separation

The Trap of Simplification

The Fractional Part Revelation

The Final Verdict

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Animated Solution for Mathematics - Functions: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation x2−4x+[x]+3=x[x], where [x] denotes the greatest integer function, has: