Question

. Solve the equation x² = [x] + {x}, where [.] is the greatest integer function and {-} is the fractional function. ​

Answer 1

Step-by-step explanation:

We have,

$\rm {x}^{2} = [x] + \{x \}$

We know that , any number can be written as the sum of its integral part and fractional part,

so,

$\rm \implies{ ([x] + \{x \})}^{2} = [x] + \{x \}$

$\rm \implies{ ([x] + \{x \})}^{2} - ([x] + \{x \}) = 0$

$\rm \implies ([x] + \{x \}) ( [x] + \{x \} - 1) = 0$

$\rm \implies [x] + \{x \} = 0 \: \: or \: \: [x] + \{x \} - 1= 0$

Now, $\rm[x] + \{x\}=0$ if x=0

And, $\rm[x] + \{x\}-1=0$

$\rm\:\implies\:\{x\}=1-[x]$,

this is possible if $\rm\:x=1$

Hence, the solition is 0 or 1