If a+b+c=0 prove that a^4+b^4+c^4/(a^2+b^2+c^2)^2=1/2
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Answer:
12
Explanation:
(a+b+c)2=a2+b2+c2+2(ab+ac+bc)=0so
ab+ac+bc=−12
(a2+b2+c2)2=a4+b4+c4+2(a2b2+a2c2+b2c2)=1so
a4+b4+c4=1−2(a2b2+a2c2+b2c2) but
(ab+ac+bc)2=a2b2+a2c2+b2c2+2(a2bc+ab2c+abc2)=14
and consequently
a2b2+a2c2+b2c2=14−2(a+b+c)abc=14
Finally
a4+b4+c4=1−24=12
12
Explanation:
(a+b+c)2=a2+b2+c2+2(ab+ac+bc)=0so
ab+ac+bc=−12
(a2+b2+c2)2=a4+b4+c4+2(a2b2+a2c2+b2c2)=1so
a4+b4+c4=1−2(a2b2+a2c2+b2c2) but
(ab+ac+bc)2=a2b2+a2c2+b2c2+2(a2bc+ab2c+abc2)=14
and consequently
a2b2+a2c2+b2c2=14−2(a+b+c)abc=14
Finally
a4+b4+c4=1−24=12
11Atulsindher11:
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