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Question Number 193410 by cortano12 last updated on 13/Jun/23

$$\{\begin{array}{l}\mathrm{x}=\sqrt{3-\sqrt{5+2\sqrt{3}}}\\ \mathrm{y}=\sqrt{3+\sqrt{5+2\sqrt{3}}}\end{array}$$$$\underbrace{\mathit{x}}$$

Answered by aba last updated on 13/Jun/23

$$\mathrm{xy}=\sqrt{9-(5+2\sqrt{3})}=\sqrt{4-2\sqrt{3}}=\sqrt{3}-1\wedge {\mathrm{x}}^2+{\mathrm{y}}^2=6$$$$\mathrm{x}+\mathrm{y}=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{xy}}=\sqrt{6+2(\sqrt{3}-1)}=\sqrt{4+2\sqrt{3}}=\sqrt{3}+1$$

Answered by Tinku Tara last updated on 13/Jun/23

$${(x+y)}^2=x^2+y^2+2xy$$$$=6+2\sqrt{9-5-2\sqrt{3}}$$$$=6+2\sqrt{4-2\sqrt{3}}(4-2\sqrt{3}={(\sqrt{3}-\sqrt{1})}^2)$$$$=6+2(\sqrt{3}-1)$$$$=4+2\sqrt{3}$$$$={(\sqrt{3}+\sqrt{1})}^2$$$$x+y=1+\sqrt{3}$$

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