Question

Let $\left[t\right]$ denote the greatest integer $\le t$ Then the equation in $x$, ${\left[x\right]}^2+2[x+2]-7=0$ has:

• A. exactly four integral solutions.
• B. infinitely many solutions.
• C. no integral solution.
• D. exactly two solutions

Answer 1

The correct option is B

infinitely many solutions.

Explanation for the correct option:

The given equation,

${\left[x\right]}^2+2[x+2]-7=0$

$$\begin{gathered} \Rightarrow {\left[x\right]}^2+2\left[x\right]+4-3=0 \\ \Rightarrow {\left[x\right]}^2+2\left[x\right]-3=0 \end{gathered}$$

Let $\left[x\right]=y$,

so,

$$\begin{gathered} y^2+2y-3=0 \\ \Rightarrow y^2+3y-y-3=0 \\ \Rightarrow (y-1)(y+3)=0 \\ \Rightarrow y=1ory=-3 \\ \Rightarrow \left[x\right]=1or\left[x\right]=-3 \end{gathered}$$

Thus, $x\in [1,2)orx\in [-3,-2)$

So, there are infinitely many solutions.

Hence, the correct option is (B)