a-a-x-a-a-x-2x-this-is-the-solution-Sir-Aifour-and-me-found-trivial-solution-a-x-0-a-x-R-x-2-8-r-r-2-4-2-4a-1-r-2-r-r-2-4-with-r-2-4a-3-3-sin-1-

Question Number 61490 by MJS last updated on 03/Jun/19

$$\sqrt{{a}−\sqrt{{a}+{x}}}+\sqrt{{a}+\sqrt{{a}−{x}}}=\mathrm{2}{x} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{Sir}\:\mathrm{Aifour}\:\mathrm{and}\:\mathrm{me}\:\mathrm{found} \\ $$$$ \\ $$$$\mathrm{trivial}\:\mathrm{solution}\:{a}={x}=\mathrm{0} \\ $$$$ \\ $$$${a},\:{x}\:\in\mathbb{R} \\ $$$$ \\ $$$${x}=\frac{\sqrt{\mathrm{2}}}{\mathrm{8}}\left({r}+\sqrt{{r}^{\mathrm{2}} +\mathrm{4}}\right)\sqrt{\mathrm{2}\left(\mathrm{4}{a}−\mathrm{1}\right)−{r}^{\mathrm{2}} −{r}\sqrt{{r}^{\mathrm{2}} +\mathrm{4}}} \\ $$$$\mathrm{with} \\ $$$${r}=−\mathrm{2}\sqrt{\frac{\mathrm{4}{a}−\mathrm{3}}{\mathrm{3}}}\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{3}}\mathrm{arcsin}\:\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\left(\mathrm{4}{a}−\mathrm{3}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\right) \\ $$$$\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{a}<{a}_{\mathrm{0}} \:\mathrm{with}\:{a}_{\mathrm{0}} \approx\mathrm{1}.\mathrm{509830340886} \\ $$

Commented by MJS last updated on 03/Jun/19

Commented by maxmathsup by imad last updated on 03/Jun/19

$${thanks}\:{sir}\:{mjs}\:{and}\:{sir}\:{ajfour}… \\ $$

Commented by MJS last updated on 03/Jun/19

The path had been found by Sir Aifour, I added some odds and ends. So it's an Indian-Austrian team work ��

Commented by behi83417@gmail.com last updated on 03/Jun/19

great job done by:MJS &Ajfour. congratulation to you two and forum!

$$\mathrm{great}\:\mathrm{job}\:\mathrm{done}\:\mathrm{by}:\mathrm{MJS}\:\&\mathrm{Ajfour}. \\ $$$$\mathrm{congratulation}\:\mathrm{to}\:\mathrm{you}\:\mathrm{two}\:\mathrm{and}\:\mathrm{forum}! \\ $$

Commented by Rasheed.Sindhi last updated on 08/Jun/19