Cho x, y, z > 0 thoả mãn: \(xy+yz+zx=6xyz\). Tìm giá trị nhỏ nhất của \(T=x\sqrt{\dfrac{x}{1+x^3}}+y\sqrt{\dfrac{y}{1+y^3}}+z\sqrt{\dfrac{z}{1+z^3}}\)

Giải hệ phương trình:

1. \(\left\{{}\begin{matrix}x+3=2\sqrt{\left(3y-x\right)\left(y+1\right)}\\\sqrt{3y-2}-\sqrt{\dfrac{x+5}{2}}=xy-2y-2\end{matrix}\right.\)

2. \(\left\{{}\begin{matrix}\sqrt{2y^2-7y+10-x\left(y+3\right)}+\sqrt{y+1}=x+1\\\sqrt{y+1}+\dfrac{3}{x+1}=x+2y\end{matrix}\right.\)

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

4. \(\left\{{}\begin{matrix}9\sqrt{\dfrac{41}{2}\left(x^2+\dfrac{1}{2x+y}\right)}=3+40x\\x^2+5xy+6y=4y^2+9x+9\end{matrix}\right.\)

5. \(\left\{{}\begin{matrix}\sqrt{xy+\left(x-y\right)\left(\sqrt{xy}-2\right)}+\sqrt{x}=y+\sqrt{y}\\\left(x+1\right)\left[y+\sqrt{xy}+x\left(1-x\right)\right]=4\end{matrix}\right.\)

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

7. \(\left\{{}\begin{matrix}x^3-12z^2+48z-64=0\\y^3-12x^2+48x-64=0\\z^3-12y^2+48y-64=0\end{matrix}\right.\)

Giải phương trình:

1. \(x^2+3x+8=\left(x+5\right)\sqrt{x^2+x+2}\)

2. \(10x^2-9x-8x\sqrt{2x^2-3x+1}+3=0\)

3. \(x^3+6x^2-2x+3-\left(5x-1\right)\sqrt{x^3+3}=0\)

4. \(4\sqrt{x+1}-1=3x+2\sqrt{1-x}+\sqrt{1-x^2}\)

5. \(4\sqrt{x+3}=1+4x+\dfrac{2}{x}\)

Giải phương trình:

1. \(\sqrt{2x^2+4x+7}=x^4+4x^3+3x^2-2x-7\)

2. \(\dfrac{4}{x}+\sqrt{x-\dfrac{1}{x}}=x+\sqrt{2x-\dfrac{5}{x}}\)

3. \(\dfrac{6-2x}{\sqrt{5-x}}+\dfrac{6+2x}{\sqrt{5+x}}=\dfrac{8}{3}\)

4. \(x^2+1-\left(x+1\right)\sqrt{x^2-2x+3}=0\)

5. \(2\sqrt{2x+4}+4\sqrt{2-x}=\sqrt{9x^2+16}\)

6. \(\left(2x+7\right)\sqrt{2x+7}=x^2+9x+7\)

Giải phương trình:

1, \(\sqrt{x^2+2x}+\sqrt{2x-1}=\sqrt{3x^2+4x+1}\)

2, \(x^3-3x^2+2\sqrt{\left(x+2\right)^3}-6x=0\)

3, \(2x^3-x^2-3x+1=\sqrt{x^5+x^4+1}\)

4, \(5\sqrt{x^4+8x}=4x^2+8\)

5, \(\left(x^2+4\right)\sqrt{2x+4}=3x^2+6x-4\)

6, \(\left(x^2-6x+11\right)\sqrt{x^2-x+1}=2\left(x^2-4x+7\right)\sqrt{x-2}\)