Subjective Type

If $$a + b + c = 0$$, prove that identities $$2(a^{4} + b^{4} + c^{4}) = (a^{2} + b^{2} + c^{2})^{2}$$.

Solution

$$2(a^4+b^4+c^4)=2((a^2+b^2+c^2)^2-2(a^2b^2+b^2c^2+c^2a^2))$$
$$=2((a^2+b^2+c^2)^2-2((ab+bc+ca)^2-2abc(a+b+c))$$
we know that $$a+b+c=0$$
so, equation deduces to
$$ = 2((a^2+b^2+c^2)^2-2(ab+bc+ca)^2)$$
$$a+b+c=0, a^2+b^2+c^2+2(ab+bc+ca)=0$$
$$ab+bc+ca=\dfrac{-(a^2+b^2+c^2)}{2}$$
substitutung this value in the expression, we get
$$= 2((a^2+b^2+c^2)^2-2\dfrac{(a^2+b^2+c^2)^2}{4})$$
$$=(a^2+b^2+c^2)^2$$

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