Question

GPT: \(1+x-2x^2=\sqrt{4x^2-1}-\sqrt{2x+1}\)

Answer 1

ĐK: 4x² - 1 > = 0 ; 2x + 1 > = 0  <=> x >=  -1/2 và x >= 1/2

PT <=> \(2+2x-4x^2=2\left(\sqrt{4x^2-1}-\sqrt{2x+1}\right)\)

<=> \(\left(\sqrt{2x+1}\right)^2-\left(\sqrt{4x^2-1}\right)^2=2\left(\sqrt{4x^2-1}-\sqrt{2x+1}\right)\)

<=> \(\left(\sqrt{2x+1}-\sqrt{4x^2-1}\right).\left(\sqrt{4x^2-1}+\sqrt{2x+1}\right)=2\left(\sqrt{4x^2-1}-\sqrt{2x+1}\right)\)

<=> \(\left(\sqrt{2x+1}-\sqrt{4x^2-1}\right).\left(\sqrt{4x^2-1}+\sqrt{2x+1}+2\right)=0\)

<=> \(\sqrt{2x+1}-\sqrt{4x^2-1}=0\left(Vì\sqrt{4x^2-1}+\sqrt{2x+1}+2>0\right)\)

<=> 2x + 1 = 4x² - 1

<=> (2x + 1).(1 - 2x + 1) = 0 <=> x = -1/2 hoặc x = 1 (chọn)

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