Is it true

Algebra Level 3

{ a + b + c = 0 a 2 b c + b 2 c a + c 2 a b = − 3 \large \begin{cases} a+b+c = 0 \\ \dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab} = -3 \end{cases}

Does the system of equations above always hold true?

No Yes a + b + c = 0 a + b + c = 0 is wrong

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Viki Zeta
May 30, 2017

a + b + c = 0 a 2 b c + b 2 a c + c 2 a b = a 3 + b 3 + c 3 a b c = 3 a b c a b c Using identity, a 3 + b 3 + c 3 = 3 a b c if a + b + c = 0 = 3 Provided that, a b c ≠ 0 a + b + c = 0 \\ \dfrac{a^2}{bc} + \dfrac{b^2}{ac} + \dfrac{c^2}{ab} \\ = \dfrac{a^3 + b^3 + c^3}{abc} \\ = \dfrac{3abc}{abc} ~~ \boxed{\text{Using identity, }a^3 + b^3 + c^3 = 3abc \text{ if a + b + c = 0}}\\ = 3 ~~ \boxed{\text{Provided that, }abc \ne 0}

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