e/ \(\sqrt{x-2}+\sqrt{6-x}=\sqrt{x^2-8x+24}\)
\(\Leftrightarrow4+2\sqrt{\left(x-2\right)\left(6-x\right)}=x^2-8x+24\)
\(\Leftrightarrow2\sqrt{-x^2+8x-12}=x^2-8x+20\)
Đặt \(\sqrt{-x^2+8x-12}=a\left(a\ge0\right)\)thì pt thành
\(2a=-a^2+8\)
\(\Leftrightarrow a^2+2a-8=0\)
\(\Leftrightarrow\orbr{\begin{cases}a=-4\left(l\right)\\a=2\end{cases}}\)
\(\Leftrightarrow\sqrt{-x^2+8x-12}=2\)
\(\Leftrightarrow-x^2+8x-12=4\)
\(\Leftrightarrow\left(x-4\right)^2=0\Leftrightarrow x=4\)
a/ \(4x^2+3x+3-4x\sqrt{x+3}-2\sqrt{2x-1}=0\)
\(\Leftrightarrow\left(4x^2-4x\sqrt{x+3}+x+3\right)+\left(2x-1-2\sqrt{2x-1}+1\right)=0\)
\(\Leftrightarrow\left(2x-\sqrt{x+3}\right)^2+\left(1-\sqrt{2x-1}\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}2x=\sqrt{x+3}\\1=\sqrt{2x-1}\end{cases}\Leftrightarrow}x=1\)
b/ \(2x-8\sqrt{2x-3}+9=0\)
\(\Leftrightarrow\left(2x-3-2.4.\sqrt{2x-3}+16\right)-4=0\)
\(\Leftrightarrow\left(4-\sqrt{2x-3}\right)^2-4=\)
\(\Leftrightarrow\left(2-\sqrt{2x-3}\right)\left(6-\sqrt{2x-3}\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}2=\sqrt{2x-3}\\6=\sqrt{2x-3}\end{cases}\Leftrightarrow\orbr{\begin{cases}x=\frac{7}{2}\\x=\frac{39}{2}\end{cases}}}\)
c/ Điều kiện \(\hept{\begin{cases}x-2\ge0\\y+2000\ge0\\z-2001\ge0\end{cases}\Leftrightarrow\hept{\begin{cases}x\ge2\\y\ge-2000\\z\ge2001\end{cases}}}\)
\(\sqrt{x-2}+\sqrt{y+2000}+\sqrt{z-2001}=\frac{1}{2}.\left(x+y+z\right)\)
\(\Leftrightarrow-2\sqrt{x-2}-2\sqrt{y+2000}-2\sqrt{z-2001}+\left(x+y+z\right)=0\)
\(\Leftrightarrow\left(x-2-2\sqrt{x-2}+1\right)+\left(y+2000-2\sqrt{y+2000}+1\right)+\left(z-2001-2\sqrt{z-2001}+1\right)=0\)
\(\Leftrightarrow\left(1-\sqrt{x-2}\right)^2+\left(1-\sqrt{y+2000}\right)^2+\left(1-\sqrt{z-2001}\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}1=\sqrt{x-2}\\1=\sqrt{y+2000}\\1=\sqrt{z-2001}\end{cases}\Leftrightarrow\hept{\begin{cases}x=3\\y=-1999\\z=2002\end{cases}}}\)
c/ \(x+y+z+23=4\sqrt{x-1}+6\sqrt{y-2}+8\sqrt{z-3}\)
\(\Leftrightarrow x+y+z+23-4\sqrt{x-1}-6\sqrt{y-2}-8\sqrt{z-3}=0\)
\(\Leftrightarrow\left(x-1-4\sqrt{x-1}+4\right)+\left(y-2-6\sqrt{y-2}+9\right)+\left(z-3-8\sqrt{z-3}+16\right)=0\)
\(\Leftrightarrow\left(2-\sqrt{x-1}\right)^2+\left(3-\sqrt{y-2}\right)^2+\left(4-\sqrt{z-3}\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}2=\sqrt{x-1}\\3=\sqrt{y-2}\\4=\sqrt{z-3}\end{cases}\Leftrightarrow\hept{\begin{cases}x=5\\y=11\\z=19\end{cases}}}\)
