1.
ĐKXĐ: \(x^2\le17\)
Đặt \(\sqrt{17-x^2}=y\) ta được hệ:
\(\left\{{}\begin{matrix}x+y+xy=9\\x^2+y^2=17\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2\left(x+y\right)+2xy=18\\x^2+y^2=17\end{matrix}\right.\)
Cộng vế:
\(\Rightarrow\left(x+y\right)^2+2\left(x+y\right)=35\)
\(\Leftrightarrow\left(x+y+1\right)^2=36\Rightarrow\left[{}\begin{matrix}x+y=5\\x+y=-7\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}y=5-x\\y=-7-x\end{matrix}\right.\)
Thế vào \(x^2+y^2=17\Rightarrow\left[{}\begin{matrix}x^2+\left(5-x\right)^2=17\\x^2+\left(-7-x\right)^2=17\end{matrix}\right.\)
\(\Leftrightarrow...\)
b.
ĐKXĐ: \(x\ge\dfrac{1}{3}\)
\(\Leftrightarrow4x^2+4+4\sqrt{3x-1}=16x\)
\(\Leftrightarrow4x^2-4x+1=4\left(3x-1\right)-4\sqrt{3x-1}+1\)
\(\Leftrightarrow\left(2x-1\right)^2=\left(2\sqrt{3x-1}-1\right)^2\)
\(\Rightarrow\left[{}\begin{matrix}2x-1=2\sqrt{3x-1}-1\\2x-1=1-2\sqrt{3x-1}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\sqrt{3x-1}\\1-x=\sqrt{3x-1}\left(x\le1\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2=3x-1\\x^2-2x+1=3x-1\end{matrix}\right.\)
\(\Leftrightarrow...\)
c.
Đặt \(\sqrt{x^2+1}=t>0\)
\(\Rightarrow t^2+3x=\left(x+3\right)t\)
\(\Leftrightarrow t^2-tx+3x-3t=0\)
\(\Leftrightarrow t\left(t-x\right)-3\left(t-x\right)=0\)
\(\Leftrightarrow\left(t-3\right)\left(t-x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}t=3\\x=t\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+1}=3\\x=\sqrt{x^2+1}\end{matrix}\right.\)
\(\Leftrightarrow...\)
