\(\sqrt{x^2-x+1}+\sqrt{x^2-x+4}=3\)
\(\Leftrightarrow\sqrt{x^2-x+1}=3-\sqrt{x^2-x+4}\)
\(\Leftrightarrow\left\{{}\begin{matrix}3-\sqrt{x^2-x+4}\ge0\\x^2-x+1=9+x^2-x+4-6\sqrt{x^2-x+4}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{1-\sqrt{21}}{2}\le x\le\frac{1+\sqrt{21}}{2}\\\sqrt{x^2-x+4}=2\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\frac{1-\sqrt{21}}{2}\le x\le\frac{1+\sqrt{21}}{2}\\x^2-x+4=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{1-\sqrt{21}}{2}\le x\le\frac{1+\sqrt{21}}{2}\\x\left(x-1\right)=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\left(TM\right)\end{matrix}\right.\)
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\(\sqrt{x^2-x+1}=t\Rightarrow t^2=x^2-x+1\)
\(\Rightarrow t^2+3=x^2-x+4\)
\(\Rightarrow t+\sqrt{t^2+3}=3\Leftrightarrow\sqrt{t^2+3}=3-t\)
\(\Leftrightarrow t^2+3=9-6t+t^2\left(t\le3\right)\)
\(\Leftrightarrow t=1\left(tm\right)\)
\(\Rightarrow x^2-x+1=1\Leftrightarrow\left[{}\begin{matrix}x=1\\x=0\end{matrix}\right.\)
