chứng minh a^4+b^4+c^4=2*(ab+bc+ca)^2 biết a+b+c=0
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AH
Akai Haruma
Giáo viên
21 tháng 7
Lời giải:
$a^4+b^4+c^4=(a^2+b^2+c^2)^2-2(a^2b^2+b^2c^2+c^2a^2)$
$=[(a+b+c)-2(ab+bc+ac)]^2-2(a^2b^2+b^2c^2+c^2a^2)$
$=[-2(ab+bc+ac)]^2-2(a^2b^2+b^2c^2+c^2a^2)$
$=4(ab+bc+ac)^2-2[(ab+bc+ac)^2-2abc(a+b+c)]$
$=4(ab+bc+ac)^2-2[(ab+bc+ac)^2]=2(ab+bc+ac)^2$
Ta có đpcm.
TT
1
27 tháng 7 2023
Ta có :
\(\left(a+b+c\right)=0\)
\(\Leftrightarrow\left(a+b+c\right)^2=0\)
\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=0\)
\(\Leftrightarrow a^2+b^2+c^2=-2\left(ab+bc+ca\right)\)
\(\Leftrightarrow\left(a^2+b^2+c^2\right)^2=\left[-2\left(ab+bc+ca\right)\right]^2\)
\(\Leftrightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)=4\left(a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\right)\left(1\right)\)
\(\Leftrightarrow a^4+b^4+c^4=4\left(a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)\right)-2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(\Leftrightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2\right)\left(2\right)\) (vì \(a+b+c=0\))
\(\left(1\right)+\left(2\right)\Rightarrow2\left(a^4+b^4+c^4\right)=4\left(a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\right)\)
\(\Rightarrow\left(a^4+b^4+c^4\right)=2\left(ab+bc+ca\right)^2\)
\(\Rightarrow dpcm\)
MD
1
Y
6 tháng 5 2019
+ a + b + c = 0 \(\Rightarrow\left(a+b+c\right)^2=0\)
\(\Rightarrow a^2+b^2+c^2=-2\left(ab+bc+ca\right)\)
+ \(a^4+b^4+c^4=\left(a^2+b^2+c^2\right)^2-2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(=\left[-2\left(ab+bc+ca\right)\right]^2-2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(=4\left(ab+bc+ca\right)^2-2\left[\left(ab+bc+ca\right)^2-2\left(ab^2c+a^2bc+abc^2\right)\right]\)
\(=2\left(ab+bc+ca\right)^2+4\left(ab^2c+abc^2+a^2bc\right)\)
\(=2\left(ab+bc+ca\right)^2+4abc\left(a+b+c\right)\)
\(=2\left(ab+bc+ca\right)^2\)
DT
1
22 tháng 7 2018
Ta có : \(a+b+c=0\)
\(\Rightarrow\left(a+b+c\right)^2=0\)
\(\Rightarrow a^2+b^2+c^2+2ab+2bc+2ac=0\)
\(\Rightarrow a^2+b^2+c^2+2\left(ab+bc+ac\right)=0\)
\(\Rightarrow a^2+b^2+c^2=-2\left(ab+bc+ac\right)\)
Bình phương hai vế , ta được :
\(\left(a^2+b^2+c^2\right)^2=\left[-2\left(ab+bc+ac\right)\right]^2\)
\(\Rightarrow a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2=4\left(a^2b^2+a^2c^2+b^2c^2+2a^2bc+2ab^2c+2abc^2\right)\) \(\left(1\right)\)
\(\Rightarrow a^4+b^4+c^4=4\left[a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)\right]-2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(\Rightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2\right)\) ( vì \(a+b+c=0\) ) \(\left(2\right)\)
Từ \(\left(1\right),\left(2\right)\):
\(2\left(a^4+b^4+c^4\right)=4\left(a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\right)\)
\(\Rightarrow\left(a^4+b^4+c^4\right)=2\left(ab+bc+ca\right)^2\)
NT
1
16 tháng 11 2019
Ta có : a + b + c = 0
( a + b + c )\(^2\) = 0
\(a^2+b^2+c^2+2ab+2bc+2ca=0\)
Nên : \(a^2+b^2+c^2=-2\left(ab+bc+ca\right)\)
\(\left(a^2+b^2+c^2\right)^2=4\left(ab+bc+ca\right)^2\)
\(a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2=4\left(ab+bc+ca\right)^2\)
\(a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2=4\left(a^2b^2+b^2c^2+c^2a^2+2ab^2c+2abc^2+2a^2bc\right)\)
\(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2+8ab^2c+8abc^2+8a^2bc\)
\(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2+8abc\left(b+c+a\right)\)
\(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2\)
Lại có : \(2\left(ab+bc+ca\right)^2\)
\(=2\left(a^2b^2+b^2c^2+c^2a^2+2ab^2c+2abc^2+2a^2bc\right)\)
\(=2a^2b^2+2b^2c^2+2c^2a^2+4ab^2c+4abc^2+4a^2bc\)
\(=2a^2b^2+2b^2c^2+2c^2a^2+4abc\left(b+c+a\right)\)
\(=2a^2b^2+2b^2c^2+2c^2a^2\)
Vì : \(2a^2b^2+2b^2c^2+2c^2a^2=2a^2b^2+2b^2c^2=2c^2a^2\)
Vậy \(a^4+b^4+c^4=2\left(ab+bc+ca\right)^2\)
DL
23 tháng 7 2017
Ta có : \(a+b+c=0\Leftrightarrow b+c=-a\)
\(\Rightarrow\left(b+c\right)^2=a^2\)(1)
\(\Rightarrow\left(a^2-b^2-c^2\right)^2=4b^2c^2\)
\(\Leftrightarrow a^4+b^4+c^4-2\left(a^2b^2-b^2c^2+2c^2a^2\right)=4b^2c^2\)
\(\Leftrightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
Từ (1) ta có :
\(a^4+b^4+c^4=2\left(ab+bc+ac\right)^2-4abc\left(a+b+c\right)\)
\(=2\left(ab+bc+ca\right)^2\)
Vì a + b + c = 0
Ta có đpcm
NT
14 tháng 3 2018
\(\left(ab+bc+ac\right)^2=a^2b^2+b^2c^2+c^2a^2\\ \Leftrightarrow a^2b^2+b^2c^2+a^2c^2+2\left(ab^2c+abc^2+a^2bc\right)=a^2b^2+b^2c^2+c^2a^2\\ \Leftrightarrow2\left(ab^2c+abc^2+a^2bc\right)=0\\ \Leftrightarrow abc\left(a+b+c\right)=0\left(đpcm;a+b+c=0\right)\)
26 tháng 5 2015
p giúp mk câu b đk k? Mk đọc mãi cũng không hiểu lắm câu a thì làm đk r
MS
1
21 tháng 2 2018
Ta có: \(a+b+c=0\)
\(\Rightarrow a+b=-c\)
\(\Rightarrow\left(a+b\right)^2=\left(-c\right)^2\)
\(\Leftrightarrow a^2+2ab+b^2=c^2\)
\(\Rightarrow a^2+b^2-c^2=-2ab\)
\(\Rightarrow\left(a^2+b^2-c^2\right)^2=a^4+b^4+c^4+2\left(a^2b^2-b^2c^2-c^2a^2\right)\)
\(\Rightarrow a^4+b^4+c^4=\left(-2ab\right)^2-2a^2b^2+2b^2c^2+2c^2a^2=2\left(a^2b^2+b^2c^2+c^2a^2\right)\left(đpcm\right)\)
a + b + c = 0
=> (a + b + c)2 = 0
=> a2 + b2 + c2 + 2ab + 2bc + 2ca = 0
=> a2 + b2 + c2 = -2(ab + 2bc + 2ca)
=> (a2 + b2 + c2)2 = [-2(ab + bc + ca)]2
=> a4 + b4 + c4 + 2a2b2 + 2b2c2 + 2c2a2 = 4(a2b2 + b2c2 + c2a2 + 2ab2c + 2a2bc + 2abc2)
=> a4 + b4 + c4 = 4a2b2 + 4b2c2 + 4c2a2 + 8a2bc + 8ab2c + 8abc2 - 2a2b2 - 2b2c2 - 2a2c2
=> a4 + b4 + c4 = 2a2b2 + 2b2c2 + 2c2a2 + 8abc(a + b + c)
=> a4 + b4 + c4= 2a2b2 + 2b2c2 + c2a2
=> a4 + b4 + c4 = 2a2b2 + 2b2c2 + 2c2a2 + 2abc(a + b + c) (Vì a + b + c = 0)
=> a4 + b4 + c4 = 2a2b2 + 2b2c2 + 2c2a2 + 2a2bc + 2ab2c + 2abc2
=> a4 + b4 + c4 = 2(a2b2 + b2c2 + c2a2 + a2bc + ab2c + abc2)
=> a4 + b4 + c4 = 2(ab + bc + ca)2 (đpcm)