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a, (a + b + c)^2 + (a - b - c)^2 +( b - c - a) ^2 + (c - a - b)^2
= (a + b + c)^2 + (a + b - c)^2 + (a - b - c)^2 + (a - b + c)^2
= (a + b)^2 + 2c(a + b) + c^2 + (a + b)^2 - 2c(a + b) + c^2 +
(a - b)^2 - 2c(a - b) + c^2 + (a - b)^2 + 2c(a - b) +c^2
= 2(a + b)^2 + 2c^2 + 2(a - b)^2 + 2c^2
= 2[(a + b)^2 + (a - b)^2] + 4c^2
=2(2a^2 + 2b^2) + 4c^2
= 4(a^2 + b^2 + c^2)
4) Ta có : A=(a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d)
=> (a+d)2 - (b+c)2= (a-d)2 - (c-b)2
=> a2+ d2+ 2ad - b2- c2- 2bc=a2 + d2 - 2ad - c2-b2+2bc
Rút gọn ta được: 4ad = 4bc => ad = bc =>\(\dfrac{a}{c}=\dfrac{b}{d}\)
1) a2+b2+c2+3=2(a+b+c) =>(a-1)2+(b-1)2+(c-1)2=0
=> a-1=b-1=c-1=0 => a=b=c=1 =>đpcm
\(VT=\frac{c-b}{\left(a-b\right)\left(c-a\right)}+\frac{a-c}{\left(a-b\right)\left(b-c\right)}+\frac{b-a}{\left(b-c\right)\left(c-a\right)}\)
\(=\frac{-\left(b-c\right)^2-\left(c-a\right)^2-\left(a-b\right)^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=\frac{-2a^2-2b^2-2c^2+2ab+2ac+2bc}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
\(=\frac{2ab-2ac+2bc-2b^2+2ab+2ac-2bc-2a^2-2ab+2ac+2bc-2c^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
\(=\frac{2\left(a-b\right)\left(b-c\right)+2\left(a-b\right)\left(c-a\right)+2\left(b-c\right)\left(c-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
\(=\frac{2}{c-a}+\frac{2}{b-c}+\frac{2}{a-b}\)
a) \(\left(a+b+c\right)^2\)
\(=\left(a+b+c\right)\left(a+b+c\right)\)
\(=a\left(a+b+c\right)+b\left(a+b+c\right)+c\left(a+b+c\right)\)
\(=a^2+ab+ac+ba+b^2+bc+ca+cb+c^2\)
\(=a^2+b^2+c^2+\left(ab+ba\right)+\left(ac+ca\right)+\left(bc+cb\right)\)
\(=a^2+b^2+c^2+2ab+2ac+2bc\)
b,c,d tương tự nhs b!
1.
a) \(\left(a+b+c\right)^2=\left[\left(a+b\right)+c\right]^2\)
\(=\left(a+b\right)^2+2\left(a+b\right)c+c^2\)
\(=a^2+b^2+c^2+2ab+2bc+2ac\)
b)\(\left(a+b-c\right)^2=\left[\left(a+b\right)-c\right]^2\)
\(=\left(a+b\right)^2-2\left(a+b\right)c+c^2\)
\(=a^2+b^2+c^2+2ab-2ac-2bc\)
c)\(\left(a-b+c\right)^2=\left[\left(a-b\right)+c\right]^2\)
\(=\left(a-b\right)^2+2\left(a-b\right)c+c^2\)
\(a^2+b^2+c^2+2ac+2bc-2ab\)
d)\(\left(a-b-c\right)^2=\left[a-\left(b+c\right)\right]^2\)
\(=a^2-2a\left(b+c\right)+\left(b+c\right)^2\)
\(=a^2+b^2+c^2-2ab-2ac+2bc\)
nhớ tik mik nha