Question

Rút gọn

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

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

Answer 1

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

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

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

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

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

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

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

\(=2.\left[\left(a+b\right)^2+\left(a-b\right)^2\right]+4.c^2=4.\left(a^2+b^2\right)+4.c^2\)

\(=4.\left(a^2+b^2+c^2\right)\)