Question

    \(\sqrt{x^2+8}-7x=\sqrt{x^2+3}-6\)(1)

\(\Leftrightarrow\sqrt{x^2+8}-3=7x-7+\sqrt{x^2+3}-2\)

\(\Leftrightarrow\frac{\left(\sqrt{x^2+8}-3\right)\left(\sqrt{x^2+8}+3\right)}{\left(\sqrt{x^2+8}+3\right)}=7\left(x-1\right)+\frac{\left(\sqrt{x^2+3}-2\right)\left(\sqrt{x^2+3}+2\right)}{\sqrt{x^2+3}+2}\)

\(\Leftrightarrow\frac{x^2+8-9}{\left(\sqrt{x^2+8}+3\right)}=7\left(x-1\right)+\frac{x^2-1}{\sqrt{x^2+3}+2}\)

\(\Leftrightarrow\frac{x^2-1}{\sqrt{x^2+8}+3}-7\left(x-1\right)-\frac{x^2-1}{\sqrt{x^2+3+2}}=0\)

\(\Leftrightarrow\left(x-1\right)\left(\frac{x+1}{\sqrt{x^2+8}+3}-7-\frac{x+1}{\sqrt{x^2+3}+2}\right)=0\)

\(\Leftrightarrow x-1=0\)

hay \(\frac{x+1}{\sqrt{x^2+8}+3}-7-\frac{x+1}{\sqrt{x^2+3}+2}=0\)(2)

Từ (1), có:

\(\sqrt{x^2+8}-\sqrt{x^2+3}=7x-6>0\)

\(\Leftrightarrow7x-6>0\)

\(\Leftrightarrow x>\frac{6}{7}\)

Khi đó, có:

     \(\frac{x+1}{\sqrt{x^2+8}+3}-\frac{\sqrt{x+1}}{\sqrt{x^2+3}+2}<0\)

\(\Rightarrow\frac{x+1}{\sqrt{x^2+8}+3}-\frac{x+1}{\sqrt{x^2+3}+2}-7<0\)

Vậy, pt (2) vô nghiệm

Do đó, pt (1) có 1 nghiệm là x = 1