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\(\left(a+b+c\right)^2+a^2+b^2+c^2\)
\(=a^2+b^2+c^2+2ab+2bc+2ca+a^2+b^2+c^2\)
\(=a^2+2ab+b^2+b^2+2bc+c^2+c^2+2ca+a^2\)
\(=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2\)
2(a-b)(c-b)+2(b-a)(c-a)+2(b-c)(a-c)
=2a^2+2b^2+2c^2-2bc-2ab-2ac
=a^2-2ac+c^2+a^2-2ab+b^2+b^2-2bc+c^2
=(a-c)^2+(a-b)^2+(b-c)^2
Ta có: (a+b+c)^2 + a^2 + b^2 + c^2
= a^2 +b^2 +c^2 + 2ab + 2ac + 2bc + a^2 + b^2 + c^2
= (a^2 +2ab+ b^2) + (b^2 +2bc+ c^2) +(c^2 +2ac+ a^2 )
= (a+b)^2 +(b+c)^2 +(c+a)^2
\(\left(a^2+b^2+c^2\right)+a^2+b^2+c^2\)
\(=a^2+b^2+c^2+2ab+2bc+2ac+a^2+b^2+c^2\)
\(=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2\)
Bài 2 :
a ) \(A=\left(a+b+c\right)^2+a^2+b^2+c^2\)
\(A=a^2+b^2+c^2+2ab+2ac+2bc+a^2+b^2+c^2\)
\(A=\left(a^2+2ab+b^2\right)+\left(a^2+2ac+c^2\right)+\left(b^2+2bc+c^2\right)\)
\(A=\left(a+b\right)^2+\left(a+c\right)^2+\left(b+c\right)^2\)
a)\(\left[\left(a-b\right)^2-2\left(a-b\right)\left(c-b\right)+\left(c-b\right)^2\right]-\left(a-b\right)^2-\left(b-c\right)^2=\left(a-b-c+b\right)^2-\left(a-b\right)^2-\left(b-c\right)^2\)
\(=\left(a-c\right)^2-\left(a-b\right)^2-\left(b-c\right)^2\) tương tự thì
A= \(\left(a-c\right)^2-\left(a-b\right)^2-\left(b-c\right)^2+\left(b-c\right)^2-\left(b-a\right)^2-\left(c-a\right)^2+\left(b-a\right)^2-\left(b-c\right)^2-\left(a-c\right)^2\)
\(=\left(a-c\right)^2-\left(a-b\right)^2-\left(b-c\right)^2+\left(b-c\right)^2-\left(a-b\right)^2-\left(a-c\right)^2+\left(a-b\right)^2-\left(b-c\right)^2-\left(a-c\right)^2\)
\(=-\left[\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2\right]\)
=a^2+b^2+c^2=2ab+2bc+2ca+a^2+b^2+c^2
=(a^2+2ab+b^2)+(b^2+2bc+c^2)+(c^2+2ca+c^2)
=(a+b)^2+(b+c)^2+(c+b)^2
Viết biểu thức dưới đây dưới dạng lập phương của 1 tổng hoặc 1 hiệu
( a +b-c)² + ( a-b+c)² -2 .(b-c)²
\(=\left(a+b-c\right)^2+2\left(a+b-c\right)\left(a-b+c\right)+\left(a-b+c\right)^2-2\left(a+b-c\right)\left(a-b+c\right)-2\left(b-c\right)^2\\ =\left(a+b-c+a-b+c\right)^2-2\left[a^2-\left(b-c\right)^2\right]-2\left(b-c\right)^2\\ =\left(2a\right)^2-2a^2+2\left(b-c\right)^2-2\left(b-c\right)^2\\ =4a^2-2a^2=2a^2\)
a,(a+b+c)^2+a^2+b^2+c^2
=a^2+b^2+c^2+2ab+2ac+2bc+a^2+b^2+c^2
=(a^2+2ab+b^2)+(b^2+2bc+c^2)+(a^2+2ac+c^2)
=(a+b)^2+(b+c)^2+(a+c)^2
b,(2a-b)(c-b)+2(b-a)(c-a)+2(b-c)(a-c)
=2(a-b)(c-b-c+a)+2(b-c)(c-a)
=2(a-b)(a-b)+2(b-c)(c-a)
=2(a-b)^2+2(b-c)(c-a)
=2(a^2-2ab+b^2)+(ab-bc-ca+c^2)
=2(a^2+b^2+c^2-ab-bc-ca)
=(a^2-2ab+b^2)+(b^2-2bc+c^2)+(c^2-2ca+a^2)
=(a-b)^2+(b-c)^2+(c-a)^2
chúc bạn học tốt!!!
\(2\left(a-b\right)\left(c-b\right)+2\left(b-a\right)\left(c-a\right)+2\left(b-c\right)\left(a-c\right)\)
\(=2\left(ac-ab-bc+b^2\right)+2\left(bc-ab-ac+a^2\right)+2\left(ab-bc-ac+c^2\right)\)
\(=2ac-2ab-2bc+2b^2+2bc-2ab-2ac+2a^2+2ab-2bc-2ac+2c^2\)
\(=2a^2+2b^2+2c^2-2ab-2bc-2ac\)
\(=\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(a^2-2ac+c^2\right)\)
\(=\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2\)