Let [t] denote the greatest integer
Question:

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

1. (1) exactly two solutions

2. (2) exactly four integral solutions

3. (3) no integral solution

4. (4) infinitely many solutions

Correct Option: , 4

Solution:

The given equation

$[x]^{2}+2[x]+4-7=0$

$\Rightarrow[x]^{2}+2[x]-3=0$

$\Rightarrow[x]^{2}+3[x]-[x]-3=0$

$\Rightarrow([x]+3)([x]-1)=0 \Rightarrow[x]=1$ or $-3$

$\Rightarrow x \in[-3,-2) \cup[1,2)$

$\therefore$ The equation has infinitely many solutions.