\(\sqrt{x-1}+7\sqrt{6-x}=15\)
\(\Leftrightarrow x-1+14\sqrt{\left(x-1\right)\left(6-x\right)}+49\left(6-x\right)=225\)
\(\Leftrightarrow14\sqrt{\left(x-1\right)\left(6-x\right)}=226-x-49\left(6-x\right)\)
\(\Leftrightarrow196\left(x-1\right)\left(6-x\right)=2304x^2-6528x+4624\)
\(\Leftrightarrow1372x-196x^2-1176-2304x^2+6528x-4624=0\)
\(\Leftrightarrow7900x-2500x^2-5800=0\)
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Đặt \(\sqrt{x-1}=a\ge0;\sqrt{6-x}=b\ge0\left(1\le x\le6\right)\)
Ta có: \(\left\{{}\begin{matrix}a+7b=15\\a^2+b^2=5\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}a=15-7b\\\left(15-7b\right)^2+b^2=5\end{matrix}\right.\)
\(\Leftrightarrow50b^2-210b+220=0\Leftrightarrow5b^2-21b+22=0\)
\(\Leftrightarrow\left(5b-11\right)\left(b-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}b=2\Leftrightarrow a=1\\b=\frac{11}{5}\Leftrightarrow a=-\frac{2}{5}\left(l\right)\end{matrix}\right.\Leftrightarrow x=2\) (t/m đk)
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