a, ĐKXĐ : \(\left[{}\begin{matrix}x\le-3\\x\ge0\end{matrix}\right.\)

TH1 : \(x\le-3\) ( LĐ )

TH2 : \(x\ge0\)

BPT \(\Leftrightarrow x^2+2x+x^2+3x+2\sqrt{\left(x^2+2x\right)\left(x^2+3x\right)}\ge4x^2\)

\(\Leftrightarrow\sqrt{\left(x^2+2x\right)\left(x^2+3x\right)}\ge x^2-\dfrac{5}{2}x\)

\(\Leftrightarrow2\sqrt{\left(x+2\right)\left(x+3\right)}\ge2x-5\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x< \dfrac{5}{2}\\x\ge-2\end{matrix}\right.\\\left\{{}\begin{matrix}x\ge\dfrac{5}{2}\\4x^2+20x+24\ge4x^2-20x+25\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}0\le x< \dfrac{5}{2}\\x\ge\dfrac{5}{2}\end{matrix}\right.\)

\(\Leftrightarrow x\ge0\)

Vậy \(S=R/\left(-3;0\right)\)

a/ \(\left\{{}\begin{matrix}\left(2x+y\right)^2-5\left(4x^2-y^2\right)+6\left(2x-y\right)^2=0\\2x+y+\dfrac{1}{2x-y}=3\end{matrix}\right.\)

Đặt \(\left\{{}\begin{matrix}2x+y=a\\2x-y=b\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}a^2-5ab+6b^2=0\left(1\right)\\a+\dfrac{1}{b}=3\left(2\right)\end{matrix}\right.\)

\(\Rightarrow\left(1\right)\Leftrightarrow\left(2b-a\right)\left(3b-a\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}a=2b\\a=3b\end{matrix}\right.\)

Thế vô (2) làm tiếp sẽ ra

b/ \(\left\{{}\begin{matrix}2x^3+y\left(x+1\right)=4x^2\left(1\right)\\5x^4-4x^6=y^2\left(2\right)\end{matrix}\right.\)

\(\Rightarrow\left(1\right)\Leftrightarrow2x^3+y=4x^2-xy\)

\(\Leftrightarrow4x^6+4x^3y+y^2=16x^4-8x^3y+x^2y^2\)

\(\Leftrightarrow4x^6+4x^3y+5x^4-4x^6=16x^4-8x^3y+x^2y^2\)

\(\Leftrightarrow11x^4-12x^3y+x^2y^2=0\)

\(\Leftrightarrow x^2\left(11x^2-12xy+y^2\right)=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\11x^2-12xy+y^2=0\end{matrix}\right.\)

Tới đây thì đơn giản rồi làm nốt nhé.

b.

Với \(x=0\) không phải nghiệm

Với \(x\ne0\) hệ tương đương:

\(\left\{{}\begin{matrix}\dfrac{y}{x^2}+\dfrac{y^2}{x}=-6\\\dfrac{1}{x^3}+y^3=19\end{matrix}\right.\)

Đặt \(\left(\dfrac{1}{x};y\right)=\left(u;v\right)\) ta được: \(\left\{{}\begin{matrix}uv^2+u^2v=-6\\u^3+v^3=19\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}3uv^2+3u^2v=-18\\u^3+v^3+19\end{matrix}\right.\)

Cộng vế với vế:

\(\left(u+v\right)^3=1\Rightarrow u+v=1\)

Thay vào \(u^2v+uv^2=-6\Rightarrow uv=-6\)

Theo Viet đảo, u và v là nghiệm của:

\(t^2-t-6=0\) \(\Rightarrow\left[{}\begin{matrix}t=-2\\t=3\end{matrix}\right.\) \(\Rightarrow\left(u;v\right)=\left(-2;3\right);\left(3;-2\right)\)

\(\Rightarrow\left(\dfrac{1}{x};y\right)=\left(-2;3\right);\left(3;-2\right)\)

\(\Rightarrow\left(x;y\right)=\left(-\dfrac{1}{2};3\right);\left(\dfrac{1}{3};-2\right)\)

a.

ĐKXĐ: \(x\ne3\)

- Với \(x\ge0\) pt trở thành:

\(\dfrac{x^2-x-12}{x-3}=2x\Rightarrow x^2-x-12=2x^2-6x\)

\(\Leftrightarrow x^2-5x+12=0\) (vô nghiệm)

- Với \(x< 0\) pt trở thành:

\(\dfrac{x^2+x-12}{x-3}=2x\Rightarrow\dfrac{\left(x-3\right)\left(x+4\right)}{x-3}=2x\)

\(\Rightarrow x+4=2x\Rightarrow x=4>0\) (ktm)

Vậy pt đã cho vô nghiệm

\(\sqrt{x^2-y+3}+\sqrt{y-x+1}=2\)

Xét \(pt\left(1\right)\Leftrightarrow2x^2+y^2-3xy-4x+3y+2=0\)

\(\Leftrightarrow\left(x-y-1\right)\left(2x-y-2\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}y=x-1\\y=2x-2\end{matrix}\right.\)

*)\(y=x-1\) thay vao \(pt(2)\) :

\(pt\Leftrightarrow\sqrt{x^2-x+4}=2\Leftrightarrow x^2-x=0\)

\(\Leftrightarrow x\left(x-1\right)=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}y=-1\\y=0\end{matrix}\right.\)

*)\(y=2x-2\) thay vao \(pt(2)\):

\(pt\Leftrightarrow\sqrt{x^2-2x+5}+\sqrt{x-1}=2\)

\(\Leftrightarrow\dfrac{x^2-2x+1}{\sqrt{x^2-2x+5}+2}+\sqrt{x-1}=0\)

\(\Leftrightarrow\left(x-1\right)\left(\dfrac{x-1}{\sqrt{x^2-2x+5}+2}+\dfrac{1}{\sqrt{x-1}}\right)=0\)

\(\Leftrightarrow x=1\)\(\Leftrightarrow y=0\)

sai r bạn ơi!