Question

Giải phương trình \(4^{x+\sqrt{x^2-2}}-5.2^{x-1+\sqrt{x^2-2}}-6=0\)

Answer 1

ĐKXĐ: \(x^2\ge2\)

\(\Leftrightarrow4^{x+\sqrt{x^2-2}}-\frac{5}{2}2^{x+\sqrt{x^2-2}}-6=0\)

Đặt \(2^{x+\sqrt{x^2-2}}=t>0\)

\(\Rightarrow t^2-\frac{5}{2}t-6=0\Rightarrow\left[{}\begin{matrix}t=4\\t=-\frac{3}{2}\left(l\right)\end{matrix}\right.\)

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

\(\Leftrightarrow\sqrt{x^2-2}=2-x\left(x\le2\right)\)

\(\Leftrightarrow x^2-2=x^2-4x+4\)

\(\Leftrightarrow x=\frac{3}{2}\)