\(x^2+4y^2+13-6x-8y=0\)
\(x^2-2\times x\times3+3^2-3^2+\left(2y\right)^2-2\times2y\times2+2^2-2^2+13=0\)
\(\left(x-3\right)^2+\left(2y+2\right)^2=0\)
\(\left(x-3\right)^2\ge0\)
\(\left(2y-2\right)^2\ge0\)
\(\Rightarrow\left(x-3\right)^2+\left(2y-2\right)^2=0\)
\(\Leftrightarrow\left(x-3\right)^2=\left(2y-2\right)^2=0\)
\(\Leftrightarrow x-3=2y-2=0\)
\(\Leftrightarrow x=3;y=1\)