Question

If a+b+c=0 prove that a^4+b^4+c^4/(a^2+b^2+c^2)^2=1/2

Answer 1

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