Question

The number of values of $x$ satisfying the equation $2(x^2-1)-\left[x\right]=0$ is:
where $[.]$ denotes GIF.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Given: $2(x^2-1)-\left[x\right]=0$
$\Rightarrow 2(x^2-1)=\left[x\right]$
Let $f\left(x\right)=2(x^2-1)$ and $g\left(x\right)=\left[x\right]$
For $f\left(x\right)=2(x^2-1)=2x^2-2$
The coefficient of $x^2=2>0$
And, the vertex is given as:
$(-\frac{b}{2a},-\frac{D}{4a})\equiv (-\frac{0}{4},-\frac{8}{4})\equiv (0,-2)$
Thus, we can draw the graph of $f\left(x\right)$ as:

Now, plotting the graph of $g\left(x\right)=\left[x\right]$ in the graph above, we get:

Thus, there are two points of intersection between the functions $f\left(x\right)$ and $g\left(x\right).$