Lời giải:
ĐKXĐ:......
Ta có:
\(x^2-6x+\sqrt{x^2-6x+7}=5\)
\(\Leftrightarrow x^2-6x+7+\sqrt{x^2-6x+7}=12\)
Đặt \(\sqrt{x^2-6x+7}=a(a\geq 0)\). Khi đó pt trở thành:
\(a^2+a=12\)
\(\Leftrightarrow a^2+a-12=0\)
\(\Leftrightarrow (a-3)(a+4)=0\Rightarrow a=3\) (do $a\geq 0$)
\(\Rightarrow x^2-6x+7=a^2=9\)
\(\Rightarrow x^2-6x-2=0\)
\(\Rightarrow x=3\pm \sqrt{11}\) (thỏa mãn)
Vậy........
\(ĐKXĐ:x^2-6x+7\ge0\)
ĐẶt: \(n=\sqrt{x^2-6x+7}\left(n\ge0\right)\)
\(\Leftrightarrow n^2-7=x^2-6x\)
Phương trình thành:
\(n^2-7+n=5\)
\(\Leftrightarrow n^2+n-12=0\)
\(\Leftrightarrow n^2+2\cdot n\cdot\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{4}-12=0\)
\(\Leftrightarrow\left(n+\dfrac{1}{2}\right)^2-\dfrac{49}{4}=0\)
\(\Leftrightarrow\left(n+\dfrac{1}{2}\right)^2=\dfrac{49}{4}\)
\(\Rightarrow\left[{}\begin{matrix}n+\dfrac{1}{2}=\dfrac{7}{2}\\n+\dfrac{1}{2}=-\dfrac{7}{2}\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}n=3\\n=-4\end{matrix}\right.\)(loại n=-4)
Với n=3
\(\Rightarrow\sqrt{x^2-6x+7}=3\)
\(\Leftrightarrow x^2-6x+7=9\)
\(\Leftrightarrow x^2-6x-2=0\)
\(\Leftrightarrow x^2-2\cdot x\cdot3+9-9-2=0\)
\(\Leftrightarrow\left(x-3\right)^2=11\)
\(\Rightarrow\left[{}\begin{matrix}x=\sqrt{11}+3\\x=-\sqrt{11}+3\end{matrix}\right.\)
