ĐKXĐ: \(x\ge3;y\ge1\)
\(\sqrt{x-3}-\sqrt{y-1}+\sqrt[3]{x^2+x+1}-\sqrt[3]{y^2+5y+7}=0\)
\(\Leftrightarrow\dfrac{x-y-2}{\sqrt{x-3}+\sqrt{y-1}}+\dfrac{x^2+x+1-y^2-5y-7}{\sqrt[3]{\left(x^2+x+1\right)}+\sqrt[3]{\left(x^2+x+1\right)\left(y^2+5y+7\right)}+\sqrt[3]{y^2+5y+7}}=0\)
Để cho gọn gàng, ta đặt:
\(\left\{{}\begin{matrix}\sqrt[3]{\left(x^2+x+1\right)}+\sqrt[3]{\left(x^2+x+1\right)\left(y^2+5y+7\right)}+\sqrt[3]{y^2+5y+7}=b>0\\\sqrt{x-3}+\sqrt{y-1}=a>0\end{matrix}\right.\)
\(\Leftrightarrow\dfrac{x-y-2}{a}+\dfrac{x^2-y^2-4y-4+x-y-2}{b}=0\)
\(\Leftrightarrow\dfrac{x-y-2}{a}+\dfrac{x^2-\left(y+2\right)^2+\left(x-y-2\right)}{b}=0\)
\(\Leftrightarrow\dfrac{x-y-2}{a}+\dfrac{\left(x-y-2\right)\left(x+y+3\right)}{b}=0\)
\(\Leftrightarrow\left(x-y-2\right)\left(\dfrac{1}{a}+\dfrac{x+y+3}{b}\right)=0\)
\(\Leftrightarrow x-y-2=0\) do \(\left\{{}\begin{matrix}x\ge3\\y\ge1\end{matrix}\right.\) \(\Rightarrow x+y+3>0\Rightarrow\dfrac{1}{a}+\dfrac{x+y+3}{b}>0\)
\(\Rightarrow x=y+2\)
Thay vào Q ta được:
\(Q=y^2-\left(y+2\right)^2+3\left(y+2\right)+4\sqrt{y}+4\)
\(\Rightarrow Q=-y+4\sqrt{y}+6=10-\left(y-4\sqrt{y}+4\right)=10-\left(\sqrt{y}-2\right)^2\le10\)
\(\Rightarrow Q_{max}=10\) khi \(\sqrt{y}-2=0\Rightarrow\left\{{}\begin{matrix}y=4\\x=6\end{matrix}\right.\)
