Question

[Nguyen Ha Vi]

Cho M= (x-a)(x-b) + (x-b)(x-c)+(x-c)(x-a) +x^2

Tinh M theo a,b,c    bik rang:  x=1/2a +1/2b +1/2c

Thanks trc!

Answer 1

[Hoàng Phúc]

Từ \(x=\frac{1}{2}a+\frac{1}{2}b+\frac{1}{2}c=\frac{1}{2}.\left(a+b+c\right)\Rightarrow2x=a+b+c\)

\(M=\left(x-a\right)\left(x-b\right)+\left(x-b\right)\left(x-c\right)+\left(x-c\right)\left(x-a\right)+x^2\)

\(=x^2-xb-ax+ab+x^2-xc-bx+bc+x^2-ax-cx+ac+x^2\)

\(=4x^2-2ax-2bx-2cx+ab+bc+ac\)

\(=4x^2-2x\left(a+b+c\right)+ab+bc+ca\)

Thay 2x=a+b+c,ta đc:

\(M=4x^2-2x.2x+ab+bc+ca=4x^2-4x^2+ab+bc+ca=ab+bc+ca\)