Multiple Choice Question (MCQ)

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}

  1. ×

    The value is equal to \dfrac{1}{2}

  2. The value is less than 0

  3. ×

    The value is greater than 0

  4. ×

    The value is equal to \dfrac{1}{4}

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

Steven Zheng posted 1 year ago

Answer

  1. abc=4 \implies a,b,c\ne 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

Steven Zheng posted 1 week ago

Scroll to Top