Question

giải phương trình:\(\sqrt{x-2\sqrt{x-1}}-\sqrt{x-1}=1\)

Answer 1

\(\sqrt{x-2\sqrt{x-1}}-\sqrt{x-1}=1\)

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

\(\Leftrightarrow\sqrt{\left(x-1\right)^2}-\sqrt{x-1}-1=0\)

\(\Leftrightarrow x-1-\sqrt{x-1}-1=0\) (1)

Đặt \(\sqrt{x-1}\) = t (t \(\ge0\))

pttt : t² - t - 1 =0

\(\Leftrightarrow\left(t-\dfrac{1}{2}\right)^2=\dfrac{5}{4}\) 

\(\Leftrightarrow\left[{}\begin{matrix}t=\dfrac{1-\sqrt{5}}{2}\left(ktm\right)\\t=\dfrac{1+\sqrt{5}}{2}\left(tm\right)\end{matrix}\right.\)

=> \(\sqrt{x-1}=\dfrac{1+\sqrt{5}}{2}\)

\(\Leftrightarrow x-1=\dfrac{3+\sqrt{5}}{2}\)

\(\Leftrightarrow x=\dfrac{5+\sqrt{5}}{2}\) (tm)

p/s: thử lại hộ mình nhaa