1. ĐKXĐ: \(-4\le x\le6\)
\(\Leftrightarrow-x^2+2x+24+\sqrt{-x^2+2x+24}-12=0\)
Đặt \(\sqrt{-x^2+2x+24}=t\ge0\)
\(t^2+t-12=0\Rightarrow\left[{}\begin{matrix}t=3\\t=-4\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow\sqrt{-x^2+2x+24}=3\)
\(\Leftrightarrow-x^2+2x+15=0\) (casio)
2. ĐKXĐ: \(x\ge1\)
\(\Leftrightarrow3x^2-18=8\sqrt{x^3-1}-24\)
\(\Leftrightarrow3\left(x^2+2\right)=8\sqrt{\left(x-1\right)\left(x^2+x+1\right)}\)
Đặt \(\left\{{}\begin{matrix}\sqrt{x^2+x+1}=a>0\\\sqrt{x-1}=b\ge0\end{matrix}\right.\)
\(\Rightarrow3\left(a^2-b^2\right)=8ab\)
\(\Leftrightarrow3a^2-8ab-3b^2=0\)
\(\Leftrightarrow\left(a-3b\right)\left(3a+b\right)=0\)
\(\Leftrightarrow a=3b\) (do \(3a+b>0\))
\(\Leftrightarrow\sqrt{x^2+x+1}=3\sqrt{x-1}\)
\(\Leftrightarrow x^2+x+1=9\left(x-1\right)\) (casio)
