Question

Giải HPT sau:

\(\left\{{}\begin{matrix}x+y-\sqrt{xy}=3\\\sqrt{x+1}+\sqrt{y+1}=4\end{matrix}\right.\)

Answer 1

ĐKXĐ: ...

Đặt \(\sqrt{xy}=t\ge0\Rightarrow x+y=t+3\)

Bình phương 2 vế pt dưới:

\(x+y+2+2\sqrt{xy+x+y+1}=16\)

\(\Leftrightarrow t+5+2\sqrt{t^2+t+4}=16\)

\(\Leftrightarrow2\sqrt{t^2+t+4}=11-t\) (\(t\le11\))

\(\Leftrightarrow4t^2+4t+16=121-22t+t^2\)

\(\Leftrightarrow3t^2+26t-105=0\Rightarrow\left[{}\begin{matrix}t=3\\t=-\frac{35}{3}\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}\sqrt{xy}=3\\x+y=6\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=6\\xy=9\end{matrix}\right.\)

Theo Viet đảo; x và y là nghiệm:

\(X^2-6X+9=0\Rightarrow X=3\Rightarrow x=y=3\)