\(A=\left(x+3y-5\right)^2-6xy+27\)

\(=x^2+6xy+9y^2-10x-30y+25-6xy+27\)

\(=x^2-10x+25+9y^2-30y+25+2\)

\(=\left(x-5\right)^2+\left(3y-5\right)^2+2\ge2\)

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