Question

Hãy viết các biểu thức sau dưới dạng tổng của ba bình phương:

a) (a+b+c)² + a² +b² +c²

b) 2(a-b)(c-b) + 2(b-a)(c-a) + 2(b-c)(a-c)

Answer 1

a: \(\left(a+b+c\right)^2+a^2+b^2+c^2\)

\(=2a^2+2b^2+2c^2+2ab+2bc+2ac\)

\(=\left(a^2+2ab+b^2\right)+\left(b^2+2bc+c^2\right)+\left(a^2+2ac+c^2\right)\)

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

b: \(=2\left(a-b\right)\left(c-b\right)-2\left(a-b\right)\left(c-a\right)+2\left(b-c\right)\left(a-c\right)\)

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

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

\(=2\left(a^2-2ab+b^2+ab-bc-ac+c^2\right)\)

\(=2\left(a^2+b^2+c^2-ab-bc-ac\right)\)

\(=2a^2+2b^2+2c^2-2ab-2bc-2ac\)

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