Question

CMR nếu a^2+b^2+4=ab - 2(a+b) thì a=b

Answer 1

\(a^2+b^2+4=ab-2\left(a+b\right)\)

\(\Leftrightarrow2a^2+2b^2+8=2ab-4a-4b\)

\(\Leftrightarrow\left(a^2+4a+4\right)+\left(b^2+4b+4\right)+\left(a^2-ab+b^2\right)=0\)

\(\Leftrightarrow\left(a+2\right)^2+\left(b+2\right)^2+\left(a-b\right)^2=0\)

Do \(\left(a+2\right)^2,\left(b+2\right)^2,\left(a-b\right)^2\ge0\)

\(\Rightarrow\left\{{}\begin{matrix}a+2=0\\b+2=0\\a-b=0\end{matrix}\right.\)\(\Rightarrow a=b=-2\left(đpcm\right)\)