given that ϕ,β are the roots of the equation 3x2−x−5=0 from the equation whose roots are 2ϕ−1/β,2β−1/ϕ


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Question Number 216355 by jikansamu last updated on 05/Feb/25

$g i v e n t h a t φ, β a r e t h e r o o t s o f t h e e q u a t i o n 3 x 2 - x - 5 = 0 f r o m t h e e q u a t i o n w h o s e r o o t s a r e 2 φ - 1 / β, 2 β - 1 / φ$
Commented by AntonCWX last updated on 05/Feb/25

$P l e a s e m a k e s u r e t o h a v e y o u r q u e s t i o n t y p e d n e a t l y a n d i n q u a l i t y .$
	Answered by AntonCWX last updated on 05/Feb/25

	$φ + β = - \frac{- 1}{3} = \frac{1}{3}$ $φ β = - \frac{5}{3}$ $\frac{2 φ - 1}{β} + \frac{2 β - 1}{φ}$ $= \frac{4 φ β - (φ + β)}{φ β}$ $= \frac{4 (- \frac{5}{3}) - (\frac{1}{3})}{- \frac{5}{3}}$ $= \frac{21}{5}$ $\frac{2 φ - 1}{β} \times \frac{2 β - 1}{φ}$ $= \frac{4 φ β - 2 (φ + β) + 1}{φ β}$ $= \frac{4 (- \frac{5}{3}) - 2 (\frac{1}{3}) + 1}{- \frac{5}{3}}$ $= \frac{19}{5}$ $N e w e q u a t i o n :$ $x^{2} - \frac{21}{5} x + \frac{19}{5} = 0$ $5 x^{2} - 21 x + 19 = 0$
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