\(x^2+4x+7=\left(x+4\right)\sqrt{x^2+7}\)
\(pt\Leftrightarrow\dfrac{x^2+4x+7}{x+4}=\sqrt{x^2+7}\)
\(\Leftrightarrow\dfrac{x^2+4x+7}{x+4}-4=\sqrt{x^2+7}-4\)
\(\Leftrightarrow\dfrac{x^2-9}{x+4}=\dfrac{x^2+7-16}{\sqrt{x^2+7}+4}\)
\(\Leftrightarrow\dfrac{x^2-9}{x+4}-\dfrac{x^2-9}{\sqrt{x^2+7}+4}=0\)
\(\Leftrightarrow\left(x^2-9\right)\left(\dfrac{1}{x+4}-\dfrac{1}{\sqrt{x^2+7}+4}\right)=0\)
Xét pt \(\dfrac{1}{x+4}-\dfrac{1}{\sqrt{x^2+7}+4}=0\Leftrightarrow\dfrac{1}{x+4}=\dfrac{1}{\sqrt{x^2+7}+4}\)
\(\Leftrightarrow x+4=\sqrt{x^2+7}+4\Leftrightarrow x=\sqrt{x^2+7}\)
\(\Leftrightarrow x^2=x^2+7\Leftrightarrow0=7\) (vô nghiệm)
Nên \(x^2-9=0\Leftrightarrow\left(x-3\right)\left(x+3\right)=0\)\(\Rightarrow\left[{}\begin{matrix}x=3\\x=-3\end{matrix}\right.\)
\(x^2+4x+7=\left(x+4\right)\sqrt{x^2+7}\)
\(\Leftrightarrow x^4+16x^2+49+8x^3+14x^2+56x=\left(x^2+8x+16\right)\left(x^2+7\right)\)
\(\Leftrightarrow x^4+8x^3+30x^2+56x+49=x^4+8x^3+23x^2+56x+112\)
\(\Leftrightarrow30x^2+49-23x^2-112=0\)
\(\Leftrightarrow7x^2-63=0\)
\(\Leftrightarrow7\left(x-3\right)\left(x+3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+3=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-3\\x=3\end{matrix}\right.\)
Đặt \(\sqrt{x^2+7}=a;x+4=b\)
\(a^2+4b-16=ab.\)
\(\left(a-4\right)\left(a-b+4\right)=0\)
