Question

Cho \(\dfrac{a+b}{3}\)=\(\dfrac{b+c}{5}\)=\(\dfrac{c+a}{6}\) cmr ac-4b² là số chính phương

Answer 1

\(\dfrac{a+b}{3}=\dfrac{b+c}{5}=\dfrac{c+a}{6}\\ \Leftrightarrow\left\{{}\begin{matrix}5a+5b=3b+3c\\5c+5a=6b+6c\\6a+6b=3c+3a\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5a+2b-3c=0\left(1\right)\\5a-6b-c=0\left(2\right)\\a+2b-c=0\left(3\right)\end{matrix}\right.\)

Từ \(\left(1\right)\left(2\right)\Leftrightarrow8b-4c=0\Leftrightarrow2b=c\)

Từ \(\left(1\right)\left(3\right)\Leftrightarrow4a-4c=0\Leftrightarrow a-c=0\Leftrightarrow a=c=2b\)

\(\Leftrightarrow ac-4b^2=2b.2b-4b^2=4b^2-4b^2=0\left(đpcm\right)\)