Question

Cho \(a\ne\pm b\) và a(a + b)(a + c) = b(b + c)(b + a). CMR: a + b + c = 0

Answer 1

Ta có:

\(a\left(a+b\right)\left(a+c\right)=b\left(b+c\right)\left(b+a\right)\)

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

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

\(\Leftrightarrow\left(a+b\right)\left[\left(a-b\right)\left(a+b\right)+c\left(a-b\right)\right]=0\)

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

\(\Leftrightarrow\left[{}\begin{matrix}a+b=0\\a-b=0\\a+b+c=0\end{matrix}\right.\)

Vì \(a\ne\pm b\Rightarrow a+b+c=0\) (đpcm)