ĐK: \(x\ge5\)
\(pt\Leftrightarrow\sqrt{5x^2+14x+9}=5\sqrt{x+1}+\sqrt{x^2-x-20}\)
Bình phương 2 vế, ta đc:
\(5x^2+14x+9=25x+5+x^2-x-20+10\sqrt{\left(x+1\right)\left(x^2-x-20\right)}\)
\(\Leftrightarrow5x^2+14x+9-25x-5-x^2+x+20=10\sqrt{\left(x+1\right)\left(x+4\right)\left(x-5\right)}\)
\(\Leftrightarrow4x^2-10x+4=10\sqrt{\left(x+1\right)\left(x-5\right)\left(x+4\right)}\)
\(\Leftrightarrow2x^2-5x+2=5\sqrt{\left(x^2-4x-5\right)\left(x+4\right)}\)
\(\Leftrightarrow2\left(x^2-4x-5\right)+3\left(x+4\right)=5\sqrt{\left(x^2-4x-5\right)\left(x+4\right)}\)
Đặt \(\sqrt{x^2-4x-5}=a\left(a\ge0\right);\sqrt{x+4}=b\left(b\ge3\right)\)
Khi đó,pt trở thành \(2a^2+3b^2=5ab\Leftrightarrow2a^2-2ab+3b^2-3ab=0\)
\(\Leftrightarrow2a\left(a-b\right)+3b\left(b-a\right)=0\Leftrightarrow\left(2a-3b\right)\left(a-b\right)=0\Leftrightarrow\left[{}\begin{matrix}a=b\\2a=3b\end{matrix}\right.\)
Với a=b \(\Rightarrow\sqrt{x^2-4x-5}=\sqrt{x+4}\Leftrightarrow x^2-5x-9=0\Leftrightarrow\left[{}\begin{matrix}x=\frac{5+\sqrt{61}}{2}\left(tmdk\right)\\x=\frac{5-\sqrt{61}}{2}\left(loai\right)\end{matrix}\right.\)
Với 2a=3b \(\Rightarrow2\sqrt{x^2-4x-5}=3\sqrt{x+4}\Leftrightarrow4\left(x^2-4x-5\right)=9\left(x+4\right)\)
\(\Leftrightarrow4x^2-25x-56=0\Leftrightarrow\left[{}\begin{matrix}x=8\left(tmdk\right)\\x=\frac{-7}{4}\left(loai\right)\end{matrix}\right.\)
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