Question

Tìm nghiệm của PT sau

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

Answer 1

ĐKXĐ: \(\left\{{}\begin{matrix}x\ne0\\x-\dfrac{1}{x}\ge0\end{matrix}\right.\)

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

\(\Rightarrow x^2-2x-1+x\sqrt{x-\dfrac{1}{x}}=0\)

\(\Rightarrow x-2-\dfrac{1}{x}+\sqrt{x-\dfrac{1}{x}}=0\)

Đặt \(\sqrt{x-\dfrac{1}{x}}=a\ge0\)

\(\Rightarrow a^2+a-2=0\Rightarrow\left[{}\begin{matrix}a=1\\a=-2\left(ktm\right)\end{matrix}\right.\)

\(\Rightarrow x-\dfrac{1}{x}=1\) (thỏa ĐKXĐ)

\(\Rightarrow x^2-x-1=0\)

\(\Rightarrow x=\dfrac{1\pm\sqrt{5}}{2}\)