Question

If a+b+c = 0, a^2+b^2+c^2 = 4

Find the value of a^4+b^4+c^4

Collected in the board: a+b+c=0 problems

Steven Zheng posted 8 hours ago

Answer

(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ac

0 = 4+2ab+2bc+2ac
(1)
ab+bc+ac = -2
(2)

(a^2+b^2+c^2)^2 = a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2

Then,

4^2 = a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2
(3)

Taking square of equation (2)

(ab+bc+ac )^2= 4

a^2b^2+b^2c^2+a^2c^2+2abc(a+b+c) = 4

Then

a^2b^2+b^2c^2+a^2c^2 = 4

Substituting to (3) gives

a^4+b^4+c^4 = 16 - 2\cdot 4 =8

Steven Zheng posted 8 hours ago

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