Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
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
Từ a+b+c=0 có b+c =-a
Suy ra (b+c)^2 = (-a)^2 hay b^2 + c^2 +2bc = a^2
hay b^2 + c^2 -a^2 = -2bc
Suy ra (b^2 + c^2 - a^2)^2 = (-2bc)^2
<=> b^4 + c^4 + a^4 +2b^2.c^2 - 2a^2.b^2 - 2a^2.c^2 = 4b^2.c^2
<=> a^4 + b^4 + c^4 = 2a^2.b^2 + 2b^2.c^2 + 2c^2.a^2
<=> 2(a^4 + b^4 + c^4) =a^4 + b^4 + c^4 + 2a^2.b^2 + 2b^2.c^2 + 2c^2.a^2
<=> 2(a^4 + b^4 + c^4 ) =(a^2 + b^2 + c^2) ( Đpcm)
Ta có: \(a+b+c=0\)
\(\Rightarrow\left(a+b+c\right)^2=0\)
\(a^2+b^2+c^2+2ab+2bc+2ca=0\)
\(\Rightarrow a^2+b^2+c^2=-2.\left(ab+bc+ca\right)\)
\(\Rightarrow\left(a^2+b^2+c^2\right)^2=\left[-2.\left(ab+bc+ca\right)\right]^2\)
\(a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2=4a^2b^2+4b^2c^2+4c^2a^2+4abc\left(a+b+c\right)\)
\(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2\)
\(\left(a^2+b^2+c^2\right)^2=a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2\)
\(\Rightarrow\left(a^2+b^2+c^2\right)^2=a^4+b^4+c^4+a^4+b^4+c^4\)
\(\Rightarrow\left(a^2+b^2+c^2\right)^2=2.\left(a^4+b^4+c^4\right)\)
đpcm
Tham khảo nhé~
\(a+b+c=0\)
\(\Rightarrow a^2+b^2+c^2\)= \(2ab+2bc+2ca=0\)
\(\Rightarrow a^2+b^2+c^2\)= \(-2\left(ab+bc+ca\right)\)
\(\Rightarrow\left(a^2+b^2+c^2\right)^2\)= \(\left(-2ab-2bc-2ca\right)^2\)
\(\Rightarrow a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2a^2\)= \(4a^2b^2+4b^2c^2+4c^2a^2+4abc\left(a+b+c\right)\)= \(4a^2b^2+4b^2c^2+4c^2a^2\)( Do a + b + c = 0 )
\(\Rightarrow a^4+b^4+c^4\)= \(2\left(a^2b^2+b^2c^2+c^2a^2\right)\).
Từ \(a+b+c=0\) \(\Rightarrow\left(a+b+c\right)^2=0\)
\(\Rightarrow a^2+b^2+c^2=-2\left(ab+bc+ca\right)\)
\(\Rightarrow\left(a^2+b^2+c^2\right)^2=\left[-2\left(ab+bc+ca\right)\right]^2\)
\(\Rightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)=4\left(a^2b^2+b^2c^2+c^2a^2\right)+8abc\left(a+b+c\right)\)
\(\Rightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2\right)\)(vì a+b+c=0)
\(\Rightarrow a^4+b^4+c^4+a^4+b^4+c^4=a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(\Rightarrow2\left(a^4+b^4+c^4\right)=\left(a^2+b^2+c^2\right)^2\)
Bo may la binh day k di hieu ashdbfgbgygygggydfsghuyfhdguuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu3