\(A=2x^2+9y^2-6xy-6x-12y+2024\)

\(=x^2+9y^2+4-6xy+4x-12y+x^2-10x+25+1995\)

\(=\left(x-3y+2\right)^2+\left(x-5\right)^2+1995\ge1995\)

The equality happens only when :
\(\left\{{}\begin{matrix}x-3y+2=0\\x-5=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}7-3y=0\\x=5\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=5\\y=\dfrac{7}{3}\end{matrix}\right.\)