Ta có: \(x+y+z=18\)
\(\Leftrightarrow y=18-x-z\)
Thế vô \(x^2-yz=18\) ta được
\(x^2-18z+xz+z^2-18=0\)
\(\Leftrightarrow4x^2+4xz+4z^2-72z-72=0\)
\(\Leftrightarrow\left(4z^2+4xz+x^2\right)-36\left(2z+x\right)+324+\left(3x^2+36x+108\right)-72-324-108=0\)
\(\Leftrightarrow\left(2z+x-18\right)^2+3\left(x+6\right)^2-504=0\)
\(\Leftrightarrow3\left(x+6\right)^2=504-\left(2z+x-18\right)^2\le504\)
\(\Rightarrow\left(x+6\right)^2\le168\)
\(\Rightarrow-2\sqrt{42}-6\le x\le2\sqrt{42}-6\)
\(\Rightarrow\hept{\begin{cases}a=42\\b=6\end{cases}}\)
\(\Rightarrow a+b=42+6=48\)
Ta có: \(x+y+z=18\)
\(\Leftrightarrow y=18-x-z\)
Thế vô \(x^2-yz=18\) ta được
\(x^2-18z+xz+z^2-18=0\)
\(\Leftrightarrow z^2+\left(x-18\right)z-18+x^2=0\)
Để phương trình bậc 2 theo z mà có nghiệm thì:
\(\Delta=\left(x-18\right)^2-4\left(x^2-18\right)\ge0\)
\(\Rightarrow-3x^2-36x+396\ge0\)
\(\Rightarrow\left(x+6\right)^2\le168\)
\(\Rightarrow-2\sqrt{42}-6\le x\le2\sqrt{42}-6\)
\(\Rightarrow\hept{\begin{cases}a=42\\b=6\end{cases}}\)
\(\Rightarrow a+b=42+6=48\)
