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ĐKXĐ: ...
\(y\left(y^2-5y+4\right)+y^2=\left(y^2-5y+4\right)\sqrt{x+1}+x+1\)
\(\Leftrightarrow\left(y^2-5y+4\right)\left(y-\sqrt{x+1}\right)+\left(y+\sqrt{x+1}\right)\left(y-\sqrt{x+1}\right)=0\)
\(\Leftrightarrow\left(y-\sqrt{x+1}\right)\left[\left(y-2\right)^2+\sqrt{x+1}\right]=0\)
\(\Leftrightarrow y=\sqrt{x+1}\Rightarrow y^2=x+1\)
Thế xuống pt dưới:
\(2\sqrt{x^2-3x+3}+6x-7=\left(x+1\right)\left(x-1\right)^2+x\sqrt{3x-2}\)
\(\Leftrightarrow2\left(\sqrt{x^2-3x+3}-1\right)+x\left(x-\sqrt{3x-2}\right)=x^3-7x+6\)
\(\Leftrightarrow\dfrac{2\left(x^2-3x+2\right)}{\sqrt{x^2-3x+3}+1}+\dfrac{x\left(x^2-3x+2\right)}{x+\sqrt{3x-2}}=\left(x+3\right)\left(x^2-3x+2\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2-3x+2=0\\\dfrac{2}{\sqrt{x^2-3x+3}+1}+\dfrac{x}{x+\sqrt{3x-2}}=x+3\left(1\right)\end{matrix}\right.\)
Xét (1) với \(x\ge\dfrac{3}{2}\):
\(\dfrac{2}{\sqrt{x^2-3x+3}+1}\le8-4\sqrt{3}< 1\)
\(\sqrt{3x-2}\ge0\Rightarrow\dfrac{x}{x+\sqrt{3x-2}}\le1\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{2}{\sqrt{x^2-3x+3}+1}+\dfrac{x}{x+\sqrt{3x-2}}< 2\\x+3>2\end{matrix}\right.\)
\(\Rightarrow\left(1\right)\) vô nghiệm
a/ ĐKXĐ: \(x^2+5x+2\ge0\Rightarrow x...\left(casio\right)\)
\(x^2+5x-2-3\sqrt{x^2+5x+2}=0\)
Đặt \(\sqrt{x^2+5x+2}=a\ge0\)
\(\Rightarrow a^4-4-3a=0\Rightarrow\left[{}\begin{matrix}a=-1< 0\left(l\right)\\a=4\end{matrix}\right.\)
\(\Rightarrow\sqrt{x^2+5x+2}=4\Leftrightarrow x^2+5x-14=0\Rightarrow\left[{}\begin{matrix}x=2\\x=-7\end{matrix}\right.\)
b/ \(x^2-6x+9+3x-22-\sqrt{x^2-3x+7}=0\)
\(\Leftrightarrow x^2-3x+7-\sqrt{x^2-3x+7}-20=0\)
Đặt \(\sqrt{x^2-3x+7}=a>0\)
\(a^2-a-20=0\Rightarrow\left[{}\begin{matrix}a=5\\a=-4< 0\left(l\right)\end{matrix}\right.\)
\(\Rightarrow\sqrt{x^2-3x+7}=5\Leftrightarrow x^2-3x-18=0\Rightarrow\left[{}\begin{matrix}x=-3\\x=6\end{matrix}\right.\)
c/ĐKXĐ: \(\left[{}\begin{matrix}x\ge-1\\x\le-2\end{matrix}\right.\)
\(x^2+3x+2-\sqrt{x^2+3x+2}-6=0\)
Đặt \(\sqrt{x^2+3x+2}=a\ge0\)
\(a^2-a-6=0\Rightarrow\left[{}\begin{matrix}a=-2< 0\left(l\right)\\a=3\end{matrix}\right.\)
\(\Rightarrow\sqrt{x^2+3x+2}=3\Leftrightarrow x^2+3x-7=0\Rightarrow\left[{}\begin{matrix}x=\dfrac{-3+\sqrt{37}}{2}\\x=\dfrac{-3-\sqrt{37}}{2}\end{matrix}\right.\)