1/ \(2\left(x^2-x+1\right)^2+x^3+1=\left(x+1\right)^2\)
Đặt \(\left\{{}\begin{matrix}x^2-x+1=a\\x+1=b\end{matrix}\right.\)
\(\Rightarrow2a^2+ab-b^2=0\)
\(\Leftrightarrow\left(a+b\right)\left(2a-b\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=-b\\2a=b\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x^2-x+1=-x-1\\2x^2-2x+2=x+1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+2=0\left(vn\right)\\2x^2-3x+1=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=1\\x=\frac{1}{2}\end{matrix}\right.\)
2/
ĐKXĐ: \(x>-1\)
Đặt \(\sqrt{2x+3}+\sqrt{x+1}=a>0\)
\(\Rightarrow a^2=3x+4+2\sqrt{2x^2+5x+3}\)
Phương trình trở thành:
\(a=a^2-20\Leftrightarrow a^2-a-20=0\Rightarrow\left[{}\begin{matrix}a=5\\a=-4< 0\left(l\right)\end{matrix}\right.\)
\(\Rightarrow\sqrt{2x+3}+\sqrt{x+1}=5\)
\(\Leftrightarrow\sqrt{2x+3}-3+\sqrt{x+1}-2=0\)
\(\Leftrightarrow\frac{2\left(x-3\right)}{\sqrt{2x+3}+3}+\frac{x-3}{\sqrt{x+1}+2}=0\)
\(\Leftrightarrow\left(x-3\right)\left(\frac{2}{\sqrt{2x+3}+3}+\frac{1}{\sqrt{x+1}+2}\right)=0\)
\(\Leftrightarrow x=3\)
Câu 3:
ĐKXĐ: \(x\ge-5\)
\(x^2+\sqrt{x+5}=5\)
Đặt \(\sqrt{x+5}=a\ge0\Rightarrow x+5=a^2\Rightarrow5=a^2-x\)
Phương trình trở thành:
\(x^2+a=a^2-x\)
\(\Leftrightarrow x^2-a^2+a+x=0\)
\(\Leftrightarrow\left(x-a\right)\left(x+a\right)+a+x=0\)
\(\Leftrightarrow\left(x+a\right)\left(x-a+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=-x\\a=x+1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x+5}=-x\left(x\le0\right)\\\sqrt{x+5}=x+1\left(x\ge-1\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2-x-5=0\\x^2+x-4=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\frac{1-\sqrt{21}}{2}\\x=\frac{-1+\sqrt{17}}{2}\end{matrix}\right.\)
