If a + b + c = 0

  a2 + b2 + c2 = 6 

Find a4 + b4 +c4

Given: a+b+c=0 ... (i) and a²+b²+c²=6 ... (ii)

From (i)

(a+b+c)²=0

a²+b²+c²+2(ab+bc+ac)=0

6+2(ab+bc+ac)=0 ... Using (ii)

ab+bc+ca=-3

Taking square of the above equation, we have

(ab+bc+ca)²=9

a²b²+b²c²+c²a²+2ab²c+2abc²+2a²bc=9

a²b²+b²c²+c²a²+2abc(a+b+c)=9

a²b²+b²c²+c²a²+2abc(0)=9    ... Using (i)

a²b²+b²c²+c²a²=9 ... (iii)

From (ii),

a²+b²+c²=6

(a²+b²+c²)²=36

a⁴+b⁴+c⁴+2(a²b²+b²c²+c²a²)=36

a⁴+b⁴+c⁴+2(9)=36 ... From (iii)

a⁴+b⁴+c⁴+18=36

a⁴+b⁴+c⁴=18