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

Given: ${\left[x\right]}^2+2[x+2]-7=0$
$\Rightarrow {\left[x\right]}^2+2\left[x\right]-3=0$
let $\left[x\right]=y.$ Then,
$y^2+2y-3=0$
$\Rightarrow (y-1)(y+3)=0$
$\Rightarrow \left[x\right]=1$ or $\left[x\right]=-3$
$\therefore x\in [1,2)\text{or}x\in [-3,-2)$