if a + b + c equal to zero prove that a to the power 4 + b to the power 4 + c to the power 4 equals to 2 whole into b square × c square + a square × b square equals to half a square + b square + c square whole square
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if a+b+c=0 the (a+b+c)^2 = 0
a^2+b^2+c^2+2(ab+bc+ca) = 0
a^2+b^2+c^2 = -2(ab+bc+ca) squaring on both sides
a^4+b^4+c^4+2(a^2b^2+b^2c^2+c^2a^2) = 4(a^2b^2+b^2c^2+c^2a^2+2(ab^2c+bc^2a+ca^2b))
a^4+b^4+c^4 = 4(a^2b^2+b^2c^2+c^2a^2+2(ab^2c+bc^2a+ca^2b))-2(a^2b^2+b^2c^2+c^2a^2)
= 2[2(a^2b^2+b^2c^2+c^2a^2+(ab^2c+bc^2a+ca^2b))-(a^2b^2+b^2c^2+c^2a^2)]
= 2((a^2b^2+b^2c^2+c^2a^2+(ab^2c+bc^2a+ca^2b)))
a^2+b^2+c^2+2(ab+bc+ca) = 0
a^2+b^2+c^2 = -2(ab+bc+ca) squaring on both sides
a^4+b^4+c^4+2(a^2b^2+b^2c^2+c^2a^2) = 4(a^2b^2+b^2c^2+c^2a^2+2(ab^2c+bc^2a+ca^2b))
a^4+b^4+c^4 = 4(a^2b^2+b^2c^2+c^2a^2+2(ab^2c+bc^2a+ca^2b))-2(a^2b^2+b^2c^2+c^2a^2)
= 2[2(a^2b^2+b^2c^2+c^2a^2+(ab^2c+bc^2a+ca^2b))-(a^2b^2+b^2c^2+c^2a^2)]
= 2((a^2b^2+b^2c^2+c^2a^2+(ab^2c+bc^2a+ca^2b)))
nilabh46:
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