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\(\left(1\right)\Leftrightarrow\hept{\begin{cases}2y+6y^2=x-y\sqrt{x-2y}\\y\ne0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}\left(x-2y\right)-y\sqrt{x-2y}-6y^2=0\\y\ne0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}\left(\frac{\sqrt{x-2y}}{y}\right)^2-\frac{\sqrt{x-2y}}{y}-6=0\\y\ne0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}t^2-t-6=0\\t=\frac{\sqrt{x-2y}}{y}\end{cases}}\)\(\Leftrightarrow\orbr{\begin{cases}t=\frac{\sqrt{x-2y}}{y}=3\\t=\frac{\sqrt{x-2y}}{y}=-2\end{cases}}\)
- Xét \(\sqrt{x-2y}=3y\left(x+3y\right)-\sqrt{x+3y}-2=0\)
\(\Leftrightarrow\hept{\begin{cases}t=\sqrt{x+3y}\left(t\ge0\right)\\t^2-t-2=0\end{cases}}\)\(\Leftrightarrow t=\sqrt{x+3y}=2\Rightarrow\hept{\begin{cases}x+3y=4\\\sqrt{x-2y}=3y\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x=4-3y\\\sqrt{4-5y}=3y\end{cases}}\Leftrightarrow\hept{\begin{cases}x=4-3y\\0< x\le\frac{4}{5}\\4-5y=9y^2\end{cases}}\Leftrightarrow\left(x;y\right)=\left(\frac{8}{3};\frac{4}{9}\right)\)
- Xét \(\sqrt{x-2y}=-2y\hept{\begin{cases}\sqrt{x-2y}=x+3y-2\\\sqrt{x-2y}=-2\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x=2-5y\\y< 0\\4y^2+7y-2=0\end{cases}}\Leftrightarrow\left(x;y\right)=\left(12;-2\right)\)
Vậy...
1/ ĐKXĐ: ...
\(\Leftrightarrow x=2016-2015\sqrt{x}-x\)
\(\Leftrightarrow2x+2015\sqrt{x}-2016=0\)
Đặt \(\sqrt{x}=t\ge0\)
\(\Rightarrow2t^2+2015t-2016=0\)
Nghiệm xấu kinh khủng, bạn tự giải
2. ĐKXĐ: ...
\(x^2+4x+4+4y^2-8y+4=4xy+13\)
\(\Leftrightarrow\left(x-2y\right)^2+4\left(x-2y\right)-5=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-2y=1\\x-2y=-5< 0\left(l\right)\end{matrix}\right.\) \(\Rightarrow x=2y+1\)
Thay xuống dưới:
\(\sqrt{\frac{\left(x+y\right)\left(x-2y\right)}{x-y}}+\sqrt{x+y}=\frac{2}{\sqrt{\left(x-y\right)\left(x+y\right)}}\)
\(\Leftrightarrow\left(x+y\right)\sqrt{x-2y}+\left(x+y\right)\sqrt{x-y}=2\)
\(\Leftrightarrow3y+1+\left(3y+1\right)\sqrt{y+1}=2\)
\(\Leftrightarrow6y+\left(3y+1\right)\left(\sqrt{y+1}-1\right)=0\)
\(\Leftrightarrow6y+\frac{\left(3y+1\right)y}{\sqrt{y+1}+1}=0\)
\(\Leftrightarrow y\left(6+\frac{3y+1}{\sqrt{y+1}+1}\right)=0\Rightarrow y=0\Rightarrow x=1\)
Giải HPT \(\left\{{}\begin{matrix}xy+x+y=x^2-2y^2\\x\sqrt{2y}-y\sqrt{x-1}=2x-2y\end{matrix}\right.\)
ĐKXĐ: ...
\(xy+x+y=x^2-2y^2\Leftrightarrow x^2-xy-2y^2-\left(x+y\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(x-2y\right)-\left(x+y\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(x-2y-1\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}x+y=0\left(vn\right)\\x=2y+1\end{matrix}\right.\)
\(\Rightarrow\left(2y+1\right)\sqrt{2y}-y\sqrt{2y}=2\left(2y+1\right)-2y\)
\(\Leftrightarrow\sqrt{2y}\left(y+1\right)=2\left(y+1\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}y+1=0\left(l\right)\\\sqrt{2y}=2\end{matrix}\right.\) \(\Rightarrow y=2\Rightarrow x=5\)
ĐK 0 <= x <= 2 ; y >= 0
(1) => \(x+2y-x+2\sqrt{x\left(2y-x\right)}=4y\)
<=> \(2\sqrt{2xy-x^2}=2y\Leftrightarrow2xy-x^2=y^2\Leftrightarrow y^2-2xy+x^2=0\Leftrightarrow y=x\)
Với y = x thay vào (2) ta có :
\(\sqrt[3]{y}+\sqrt{2-y}=2\)