If the roots α and β of the equation ax^2 + bx + c = 0 are real and of opposite sign then the roots of the equation α(x − β)^2 + β(x − α)^2 is/are (1) Positive (2) Negative (3) Real and opposite sign (4) Imaginary


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Question Number 20052 by Tinkutara last updated on 21/Aug/17

$If the roots α and β of the equation$ ${a x}^{2} + b x + c = 0 are real and of opposite$ $sign then the roots of the equation$ $α {(x - β)}^{2} + β {(x - α)}^{2} is / are$ $(1) Positive$ $(2) Negative$ $(3) Real and opposite sign$ $(4) Imaginary$
	Answered by ajfour last updated on 21/Aug/17

	$g i v e n α β < 0$ $s e c o n d e q u a t i o n r e f o r m e d i s$ $(α + β) x^{2} - 4 α β x + α β (α + β) = 0$ $D = 16 α^{2} β^{2} - 4 α β {(α + β)}^{2} > 0$ $f o r α β < 0$ $H e n c e r o o t s o f s e c o n d e q u a t i o n$ $γ, δ a r e r e a l .$ $γ δ = \frac{C}{A} = \frac{α β (α + β)}{α + β} = α β < 0$ $H e n c e γ a n d δ h a v e o p p o s i t e s i g n s .$ $(3) i s t h e c o r r e c t o p t i o n .$
	Commented by Tinkutara last updated on 21/Aug/17

	$Thank you very much Sir!$
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