\(\sqrt{x^2+x+25}-\sqrt{x^2+x+16}=1\)
\(pt\Leftrightarrow\sqrt{x^2+x+25}-5-\sqrt{x^2+x+16}-4=0\)
\(\Leftrightarrow\dfrac{x^2+x+25-25}{\sqrt{x^2+x+25}+5}-\dfrac{x^2+x+16-16}{\sqrt{x^2+x+16}+4}=0\)
\(\Leftrightarrow\dfrac{x^2+x}{\sqrt{x^2+x+25}+5}-\dfrac{x^2+x}{\sqrt{x^2+x+16}+4}=0\)
\(\Leftrightarrow\left(x^2+x\right)\left(\dfrac{1}{\sqrt{x^2+x+25}+5}-\dfrac{1}{\sqrt{x^2+x+16}+4}\right)=0\)
Pt \(\dfrac{1}{\sqrt{x^2+x+25}+5}-\dfrac{1}{\sqrt{x^2+x+16}+4}=0\) vô nghiệm
\(\Rightarrow x^2+x=0\Rightarrow x\left(x+1\right)=0\)\(\Rightarrow\left[{}\begin{matrix}x=0\\x=-1\end{matrix}\right.\)
đk mọi x
\(\left\{{}\begin{matrix}\sqrt{x^2+x+25}=a\\\sqrt{x^2+x+16}=b\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a\ge\dfrac{3\sqrt{11}}{2}\\b\ge\dfrac{3\sqrt{7}}{2}\\a^2-b^2=9\end{matrix}\right.\)
\(\Leftrightarrow\)\(\left\{{}\begin{matrix}a-b=1\\a+b=9\end{matrix}\right.\)
\(2a=10\Rightarrow a=5\) \(\Rightarrow x^2+x+25=25\Rightarrow\left[{}\begin{matrix}x=0\\x=-1\end{matrix}\right.\)