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Question Number 174356 by behi834171 last updated on 30/Jul/22

x^{3} + 4 x^{2} - 9 x - 14 = 0

1 . \frac{1}{x_{1}^{2}} + \frac{1}{x_{2}^{2}} + \frac{1}{x_{3}^{2}} = ?

2 . Σ \frac{x_{1}^{2} + x_{2}^{2}}{x_{3}} = ?

Answered by behi834171 last updated on 30/Jul/22

l e t : y = \frac{1}{x} \Rightarrow \frac{1}{y^{3}} + \frac{4}{y^{2}} - \frac{9}{y} - 14 = 0

\Rightarrow y^{3} + \frac{9}{14} y^{2} - \frac{2}{7} y - \frac{1}{14} = 0

Σ \frac{1}{x_{i}^{2}} = Σ y_{i}^{2} = {(Σ y)}^{2} - 2 (Σ y_{i} . y_{j}) =

= {(- \frac{9}{14})}^{2} - 2 (- \frac{2}{7}) = \frac{81}{196} + \frac{4}{7} = \frac{81 + 112}{196} =

= \frac{193}{196} .

\frac{a^{2} + b^{2}}{c} = \frac{{(a + b)}^{2} - 2 a b}{c} = \frac{{(- 4 - c)}^{2} - 2 \frac{a b c}{c}}{c} =

= \frac{16 + 8 c + c^{2} - \frac{28}{c}}{c} = \frac{16 c + 8 c^{2} + c^{3} - 28}{c^{2}} =

= \frac{16}{c} + c - \frac{28}{c^{2}} + 8 \Rightarrow

Σ \frac{x_{1}^{2} + x_{2}^{2}}{x_{3}} = 24 + Σ x_{i} + 16 Σ \frac{1}{x_{i}} - 28 Σ \frac{1}{x_{i}^{2}} =

= 24 + (- 4) + 16 (- \frac{9}{14}) - 28 (\frac{193}{196}) =

= \frac{- 28 \times 193 + 24 \times 196 - 14 \times 16 \times 9 - 4 \times 196}{196} =

= - \frac{875}{49} .

Commented by Tawa11 last updated on 31/Jul/22

Great sir

Answered by BaliramKumar last updated on 30/Jul/22

1 . \frac{1}{α^{2}} + \frac{1}{β^{2}} + \frac{1}{γ^{2}} = {(\frac{1}{α} + \frac{1}{β} + \frac{1}{γ})}^{2} - 2 (\frac{1}{α β} + \frac{1}{β γ} + \frac{1}{γ α})

= {(\frac{α β + β γ + γ α}{α β γ})}^{2} - 2 (\frac{α + β + γ}{α β γ})

= {(\frac{- 9}{14})}^{2} - 2 (\frac{- 4}{14}) = \frac{81}{196} + \frac{8}{14}

= \frac{193}{196}

Commented by behi834171 last updated on 30/Jul/22

t h a n k y o u s i r f o r y o u r t i m e .

r i g h t a n s w e r ✓ .

Commented by Tawa11 last updated on 31/Jul/22

Great sir

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