MCQ
If $[\text{x}^2]-5[\text{x}]+6=0,$ where [.] denotes the greatest integer function, then:
- A$\text{x}\in[3,4]$
- B$\text{x}\in\big(2,3\big]$
- C$\text{x}\in\big[2,3\big]$
- ✓$\text{x}\in\big[2,4\big)$
✓
Answer
Correct option: D.
$\text{x}\in\big[2,4\big)$
The given equation is $[\text{x}^2]-5[\text{x}]+6=0$
$[\text{x}^2]-5[\text{x}]+6=0$
$\Rightarrow[\text{x}^2\big]-3\big[\text{x}\big]-2\big[\text{x}\big]+6=0$
$\Rightarrow\big[\text{x}\big]\big([\text{x}]-3\big)-2\big([\text{x]}-3\big)=0$
$\Rightarrow\big([\text{x}]-2)\big([\text{x}]-3)=0$
$\Rightarrow\big[\text{x}\big]-2=0$ or $[\text{x}]-3=0$
$\Rightarrow[\text{x}]=2$ or $[\text{x}]=3$
$\Rightarrow\text{x}\in\big[2,3\big)$ or $\big[3,4\big)$
$\Rightarrow\text{x}\in\big[2,4\big)$
$[\text{x}^2]-5[\text{x}]+6=0$
$\Rightarrow[\text{x}^2\big]-3\big[\text{x}\big]-2\big[\text{x}\big]+6=0$
$\Rightarrow\big[\text{x}\big]\big([\text{x}]-3\big)-2\big([\text{x]}-3\big)=0$
$\Rightarrow\big([\text{x}]-2)\big([\text{x}]-3)=0$
$\Rightarrow\big[\text{x}\big]-2=0$ or $[\text{x}]-3=0$
$\Rightarrow[\text{x}]=2$ or $[\text{x}]=3$
$\Rightarrow\text{x}\in\big[2,3\big)$ or $\big[3,4\big)$
$\Rightarrow\text{x}\in\big[2,4\big)$
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