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Question Number 161697 by mnjuly1970 last updated on 21/Dec/21

Answered by aleks041103 last updated on 21/Dec/21

$$letn=\lfloor x\rfloor andr=x-\lfloor x\rfloor \Rightarrow 0⩽r<1$$$$\Rightarrow 0⩽\sqrt{r}<1$$$$\sqrt{2n+r}-\sqrt{r}=1$$$$\Rightarrow \sqrt{2n+r}=1+\sqrt{r}$$$$\Rightarrow 1⩽\sqrt{2n+r}<2$$$$\Rightarrow 1⩽2n+r<4$$$$Onviouslyn=1.$$$$\Rightarrow \sqrt{2+r}-\sqrt{r}=1$$$$\Rightarrow 2+r=1+r+2\sqrt{r}$$$$\Rightarrow 1=2\sqrt{r}\Rightarrow r=\frac{1}{4}$$$$\Rightarrow x=1.25$$$$\sqrt{1+1.25}-\sqrt{0.25}=1.5-0.5=1$$$$$$$$Ans.x=1.25$$$$$$

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