If real numbers $$ a,b, c $$ satisfy $$ a+b+c=0 $$ , $$ abc=4 $$ , which one best describes the value of $$ \dfrac{1}{a} +\dfrac{1}{b} +\dfrac{1}{c} $$

Question

If real numbers $$ a,b, c $$ satisfy $$ a+b+c=0 $$ , $$ abc=4 $$ , which one best describes the value of $$ \dfrac{1}{a}  +\dfrac{1}{b}  +\dfrac{1}{c}  $$

Uniteasy Admin · Accepted Answer

$$abc=4 \implies a,b,c
e 0$$, the value of the expression is not equal to 0.
$$a+b+c=0 \implies (a+b+c)^2 = 0$$
Therefore,
$$ \dfrac{1}{a}  +\dfrac{1}{b}  +\dfrac{1}{c}  $$
$$=\dfrac{bc+ac+ab}{abc}$$
$$=\dfrac{bc+ac+ab-(a+b+c)^2}{abc}$$
$$=\dfrac{-(a^2+b^2+c^2+ab+bc+ac)}{abc}  $$
$$=\dfrac{- (2a^2+2b^2+2c^2+2ab+2bc+2ac)}{2abc}  $$
$$= - \dfrac{(a+b)^2+(b+c)^2+(a+c)^2}{abc}  <0$$

Multiple Choice Question (MCQ)

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