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Question Number 66508 by Masumsiddiqui399@gmail.com last updated on 16/Aug/19

Commented by Prithwish sen last updated on 16/Aug/19

((√(7−4(√3))))^(x+(√(x+2))) =(2−(√3))^(x+^3 (√(2x+4)))   ∵7−4(√3) = (2−(√3))^2   ∴ x+(√(x+2))=x+^3 (√(2x+1))  ⇒x=2

$$\left(\sqrt{\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}}\right)^{\mathrm{x}+\sqrt{\mathrm{x}+\mathrm{2}}} =\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)^{\mathrm{x}+^{\mathrm{3}} \sqrt{\mathrm{2x}+\mathrm{4}}} \\ $$$$\because\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}\:=\:\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \\ $$$$\therefore\:\mathrm{x}+\sqrt{\mathrm{x}+\mathrm{2}}=\mathrm{x}+^{\mathrm{3}} \sqrt{\mathrm{2x}+\mathrm{1}}\:\:\Rightarrow\mathrm{x}=\mathrm{2} \\ $$

Commented by MJS last updated on 16/Aug/19

x=2∨x=−2

$${x}=\mathrm{2}\vee{x}=−\mathrm{2} \\ $$

Commented by Prithwish sen last updated on 16/Aug/19

yes sir. I have missed the negative  root.

$$\mathrm{yes}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{have}\:\mathrm{missed}\:\mathrm{the}\:\mathrm{negative}\:\:\mathrm{root}. \\ $$

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