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

Options

A. no solution
B. exactly two solutions `(–oo, oo)`
C. a unique solution in `(–oo,1)`
D. a unique solution in `(–oo,oo)`

Answer

To solve the equation \( x^2 - 4x + [x] + 3 = x[x] \), where \([x]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ x^2 - 4x + [x] + 3 = x[x] \]
Rearranging the equation gives us:
\[ x^2 - 4x + [x] + 3 - x[x] = 0 \]
This simplifies to:
\[ x^2 - x[x] - 4x + [x] + 3 = 0 \]

### Step 2: Group terms
We can group the terms involving \([x]\):
\[ x^2 - (x + 4)x + ([x] + 3) = 0 \]
This can be rearranged to:
\[ x^2 - x[x] - 4x + [x] + 3 = 0 \]

### Step 3: Factor the equation
We can factor the equation:
\[ x^2 - x[x] - 4x + [x] + 3 = 0 \]
This can be expressed as:
\[ (x - 1)(x - [x]) - 3(x - 1) = 0 \]
Factoring out \((x - 1)\):
\[ (x - 1)(x - [x] - 3) = 0 \]

### Step 4: Solve for \(x\)
Setting each factor to zero gives us:
1. \(x - 1 = 0 \Rightarrow x = 1\)
2. \(x - [x] - 3 = 0 \Rightarrow x - [x] = 3\)

### Step 5: Analyze the second equation
The equation \(x - [x] = 3\) implies that the fractional part of \(x\) is equal to 3. However, the fractional part of any number is always less than 1. Therefore, this equation has no solutions.

### Step 6: Conclusion
The only solution we found is \(x = 1\). Since there are no other solutions, we conclude that the equation has a unique solution.

Thus, the final answer is:
\[ \text{The equation has a unique solution: } x = 1. \]