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![](https://rs.olm.vn/images/avt/0.png?1311)
1,\(x^2-2y^2-xy=0\)
<=> \(\left(x-2y\right)\left(x+y\right)=0\)
<=> \(\orbr{\begin{cases}x=2y\\x=-y\end{cases}}\)
Sau đó bạn thế vào PT dưới rồi tính
3. ĐKXĐ \(x\le1\); \(x+2y+3\ge0\)
.\(2y^3-\left(x+4\right)y^2+8y+x^2-4x=0\)
<=> \(\left(2y^3-xy^2\right)+\left(x^2-4y^2\right)-\left(4x-8y\right)=0\)
<=> \(\left(x-2y\right)\left(-y^2+x+2y-4\right)=0\)
Mà \(-y^2+2y-4=-\left(y-1\right)^2-3\le-3\); \(x\le1\)nên \(-y^2+x+2y-4< 0\)
=> \(x=2y\)
Thế vào Pt còn lại ta được
\(\sqrt{\frac{1-x}{2}}+\sqrt{2x+3}=\sqrt{5}\)ĐK \(-\frac{3}{2}\le x\le1\)
<=> \(\frac{1-x}{2}+2x+3+2\sqrt{\frac{\left(1-x\right)\left(2x+3\right)}{2}}=5\)
<=> \(\sqrt{2\left(1-x\right)\left(2x+3\right)}=-\frac{3}{2}x+\frac{3}{2}\)
<=> \(\sqrt{2\left(1-x\right)\left(2x+3\right)}=-\frac{3}{2}\left(x-1\right)\)
<=> \(\orbr{\begin{cases}x=1\\\sqrt{2\left(2x+3\right)}=\frac{3}{2}\sqrt{1-x}\end{cases}}\)=> \(\orbr{\begin{cases}x=1\\x=-\frac{3}{5}\end{cases}}\)(TMĐK )
Vậy \(\left(x;y\right)=\left(1;\frac{1}{2}\right),\left(-\frac{3}{5};-\frac{3}{10}\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
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![](https://rs.olm.vn/images/avt/0.png?1311)
\(\begin{cases}x^3-6x^2y+9xy^2-4y^3=0\left(1\right)\\\sqrt{x-y}+\sqrt{x+y}=2\left(2\right)\end{cases}\)
\(\left(1\right)\Leftrightarrow x^3-2x^2y+xy^2-4y^3+8xy^2-4x^2y=0\)
\(\Leftrightarrow x\left(x^2-2xy+y^2\right)-4y\left(x^2-2xy+y^2\right)=0\)
\(\Leftrightarrow\left(x^2-2xy+y^2\right)\left(x-4y\right)=0\)
\(\Leftrightarrow\left(x-y\right)^2\left(x-4y\right)=0\)
\(\Leftrightarrow\left[\begin{array}{nghiempt}\left(x-y\right)^2=0\\x-4y=0\end{array}\right.\)\(\Leftrightarrow\left[\begin{array}{nghiempt}x=y\\x=4y\end{array}\right.\)
- Xét \(x=y\) thay vào (2) ta có:
\(\left(2\right)\Leftrightarrow\sqrt{x-x}+\sqrt{x+x}=2\)
\(\Leftrightarrow\sqrt{2x}=2\Leftrightarrow2x=4\Leftrightarrow x=2\).Mà \(\begin{cases}x=2\\x=y\end{cases}\)\(\Rightarrow x=y=2\)
- Xét \(x=4y\) thay vào (2) ta có:
\(\left(2\right)\Leftrightarrow\sqrt{4y-y}+\sqrt{4y+y}=2\)
\(\Leftrightarrow\sqrt{3y}+\sqrt{5y}=2\)\(\Leftrightarrow\sqrt{y}\left(\sqrt{5}+\sqrt{3}\right)=2\)
\(\Leftrightarrow y\left(\sqrt{15}+4\right)=2\)\(\Leftrightarrow y=\frac{2}{\sqrt{15}+4}\).Mà \(\begin{cases}x=4y\\y=\frac{2}{\sqrt{15}+4}\end{cases}\)\(\Leftrightarrow x=\frac{8}{\sqrt{15}+4}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
V1 <=> \(xy^2+4y^2+8-x^2+2x-4x=0\)
<=> \(y^2\left(x+4\right)+2\left(x+4\right)-x\left(x+4\right)=0\)
<=> \(\left(y^2+2-x\right)\left(x+4\right)=0\)
<=>\(\orbr{\begin{cases}x=y^2+2\\x=-4\end{cases}}\)
TH1: Thay \(x=y^2+2\)vào V2:
\(y^2+2+y+3=3\sqrt{2y-1}\)
<=> \(2y^2+\left(2y-1\right)-6\sqrt{2y-1}+9+2=0\)
<=> \(2\left(y^2+1\right)+\left(\sqrt{2y-1}-3\right)^2=0\)
<=> \(\hept{\begin{cases}y^2=-1\left(\text{loại}\right)\\\sqrt{2y-1}=3\end{cases}}\)
<=> 2y - 1 = 9
<=> y = 5
=> \(x=y^2+2=27\)
TH2: Thay x = -4 vào V2, tương tự đc \(\orbr{\begin{cases}y=10-3\sqrt{10}\\y=10+3\sqrt{10}\end{cases}}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có: \(\sqrt{8x-y+5}+\sqrt{x+y-1}=3\sqrt{x}+2\)
\(\Leftrightarrow8x-y+5+x+y-1+2\sqrt{\left(8x-y+5\right)\left(x+y-1\right)}=9x+12\sqrt{x}+4\)
\(\Leftrightarrow9x+4+2\sqrt{8x^2-y^2+7xy-3x+6y-5}=9x+4+12\sqrt{x}\)
\(\Leftrightarrow\sqrt{8x^2-y^2+7xy-3x+6y-5}=6\sqrt{x}\)
\(\Leftrightarrow8x^2-y^2+7xy-3x+6y-5=36x\)
\(\Leftrightarrow8x^2-y^2+7xy-39x+6y-5=0\)
\(\Leftrightarrow\left(8x^2+8xy-40x\right)-y^2-xy-5+x+6y=0\)
\(\Leftrightarrow8x\left(x+y-5\right)-\left(y^2+xy-5y\right)+\left(x+y-5\right)=0\)
\(\Leftrightarrow\left(x+y-5\right)\left(8x-y+1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}y=5-x\\y=8x+1\end{cases}}\)
Thay vào pt dưới ta có:
\(\sqrt{xy}+\frac{1}{\sqrt{x}}=\sqrt{8x-y+5}\left(1\right)\)
+) với y=5-x (1) thành:
\(\sqrt{x\left(5-x\right)}+\frac{1}{\sqrt{x}}=\sqrt{8x-\left(5-x\right)+5}\)
\(\Leftrightarrow\sqrt{5x-x^2}+\frac{1}{\sqrt{x}}=\sqrt{9x}\)\(\Leftrightarrow\sqrt{5x^2-x^3}+1=3x\)\(\Leftrightarrow\sqrt{5x^2-x^3}=3x-1\)
\(\Leftrightarrow\hept{\begin{cases}x\ge\frac{1}{3}\\5x^2-x^3=9x^2-6x+1\end{cases}\Leftrightarrow\hept{\begin{cases}x\ge\frac{1}{3}\\x^3+4x^2-6x+1=0\end{cases}\Leftrightarrow}\hept{\begin{cases}x\ge\frac{1}{3}\\x=1\left(tm\right)\end{cases}}}\)
Với x=1=>y=4
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\text{Condition}:x,y\ge0\)
\(\hept{\begin{cases}x^2+2x=4-\sqrt{y}\left(M_1\right)\\y^2+2y=4-\sqrt{x}\left(M_2\right)\end{cases}}\)
\(\left(M_1\right)-\left(M_2\right)\Leftrightarrow\left(x^2-y^2\right)+2\left(x-y\right)+\left(\sqrt{x}-\sqrt{y}\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)+2\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)+\left(\sqrt{x}-\sqrt{y}\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=y\\\left(x+y\right)\left(\sqrt{x}+\sqrt{y}\right)+2\left(\sqrt{x}+\sqrt{y}\right)+1=0\left(M_3\right)\end{cases}}\)
x=0 khong phai nghiem PT\(\Rightarrow M_3\)(fail)
Thay x=y vao
:D
a,\(\hept{\begin{cases}x^2+y^2+\frac{2xy}{x+y}=1\\\sqrt{x+y}=x^2-y\end{cases}}\)
ĐK: \(x+y\ge0\)
\(\Leftrightarrow\hept{\begin{cases}\left(x+y\right)^2-2xy+\frac{2xy}{x+y}=1\left(1\right)\\\sqrt{x+y}=x^2-y\left(2\right)\end{cases}}\)
Đặt \(\hept{\begin{cases}x+y=a\\2xy=b\end{cases}\left(a\ge0\right)}\)
\(\left(1\right)\Leftrightarrow a^2-b+\frac{b}{a}=1\)
\(\Leftrightarrow a^3-ab-a+b=0\)
\(\Leftrightarrow\left(a-1\right)\left(a^2+a-b\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}a=1\\a^2+a-b=0\end{cases}\Leftrightarrow}\orbr{\begin{cases}x+y=1\left(3\right)\\\left(x+y\right)^2+\left(x+y\right)-xy=0\left(4\right)\end{cases}}\)
Thay (3) vào (2) ta được
\(x^2-y=1\Leftrightarrow y=x^2-1\)
\(\Rightarrow1-x=x^2-1\Leftrightarrow x^2+x-2=0\Leftrightarrow\orbr{\begin{cases}x=1\Rightarrow y=0\\x=-2\Rightarrow y=3\end{cases}}\)
Giải (4)
Ta có \(\left(x+y\right)^2\ge4xy\Rightarrow\left(x+y\right)^2-xy>0\)
do đó (4) không xảy ra
Vậy..........