a: \(a^2+4b^2+9c^2=2ab+6bc+3ac\)
=>\(2a^2+8b^2+18c^2-4ab-12bc-6ac=0\)
=>\(a^2-4ab+4b^2+4b^2-12bc+9c_{}^2+a^2-6ac+9c^2=0\)
=>\(\left(a-2b\right)^2+\left(2b-3c\right)^2+\left(a-3c\right)^2=0\)
=>\(\begin{cases}a-2b=0\\ 2b-3c=0\\ 3c-a=0\end{cases}\Rightarrow a=2b=3c\)
\(A=\left(a-2b+1\right)^{2022}+\left(2b-3c-1\right)^{2023}+\left(3c-a+1\right)^{2024}\)
\(=\left(a-a+1\right)^{2022}+\left(2b-2b-1\right)^{2023}+\left(a-a+1\right)^{2024}\)
=1-1+1
=1
b: \(x^2+2xy+6x+6y+2y^2+8=0\)
=>\(x^2+2xy+y^2+6\left(x+y\right)+9+y^2-1=0\)
=>\(\left(x+y+3\right)^2-1=-y^2\)
=>\(-y^2=\left(x+y+2\right)\left(x+y+4\right)\)
=>\(-y^2=\left(x+y+2024-2022\right)\left(x+y+2024-2020\right)\)
=>\(-y^2=\left(A-2022\right)\left(A-2020\right)\)
mà \(-y^2\le0\forall y\)
nên (A-2022)(A-2020)<=0
=>2020<=A<=2022
\(A_{\min}=2020\) khi x+y+2=0 và y=0
=>y=0 và x=-2-y=-2-0=-2
\(A\max=2022\) khi x+y+4=0 và y=0
=>y=0 và x=-y-4=-4
