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Question Number 32534 by naka3546 last updated on 27/Mar/18

Answered by MJS last updated on 27/Mar/18

$$m+\sqrt{n}=k$$$$f\left(x\right)=k$$$$x^2+x-k=0$$$$x=-\frac{1}{2}\pm \frac{1}{2}\sqrt{4k+1}$$$$f\left(\frac{1}{x}\right)=\frac{3k+1}{2k^2}\pm \frac{k+1}{2k^2}\sqrt{4k+1}$$$$\mathrm{case}1:+/+$$$$\mathrm{case}2:+/-$$$$\mathrm{case}3:-/-$$$$$$$$\mathrm{case}1$$$$\frac{3k+1}{k^2}+\frac{k+1}{k^2}\sqrt{4k+1}=\frac{1}{10}$$$$(k+1)\sqrt{4k+1}=\frac{1}{10}{k}^2-3k-1$$$$k^4-460k^3-20k^2=0$$$$\mathrm{of}\mathrm{the}4\mathrm{solutions}\mathrm{only}\mathrm{one}\mathrm{is}\mathrm{valid}$$$$k=230+42\sqrt{30}$$$$\Rightarrow m=230;n=52920$$$$100m+n=75920$$$$$$$$\mathrm{case}2$$$$\frac{3k+1}{k^2}=\frac{1}{10}$$$$k^2-30k-10=0$$$$k=15-\sqrt{235}\mathrm{not}\mathrm{valid}(\sqrt{n}=-\sqrt{235})$$$$k=15+\sqrt{235}$$$$m=15;n=235$$$$100m+n=1735$$$$$$$$\mathrm{case}3$$$$\frac{3k+1}{k^2}+\frac{k+1}{k^2}\sqrt{4k+1}=-\frac{1}{10}$$$$(k+1)\sqrt{4k+1}=-\frac{1}{10}{k}^2-3k-1$$$$\mathrm{no}\mathrm{solution}$$

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