Question

giai pt

\(\log_2\left(\sqrt{x^2-x+1}+\sqrt{x+1}\right)=\log_2\left(x+2\right)\)

Answer 1

ĐKXĐ: \(x\ge-1\)

\(log_2\left(\sqrt{x^2-x+1}+\sqrt{x+1}\right)=log_2\left(x+2\right)\)

\(\Rightarrow\sqrt{x^2-x+1}+\sqrt{x+1}=x+2\)

\(\Leftrightarrow\sqrt{x^2-x+1}=x+2-\sqrt{x+1}\)

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

\(\Leftrightarrow\left(x+2\right)\sqrt{x+1}=3x+2\) \(\left(x\ge-\dfrac{2}{3}\right)\)

\(\Leftrightarrow\left(x+2\right)^2\left(x+1\right)=\left(3x+2\right)^2\)

\(\Leftrightarrow x\left(x^2-4x-4\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x=2+2\sqrt{2}\\x=2-2\sqrt{2}\left(loại\right)\end{matrix}\right.\)