α and β are roots of ax^2 +bx +c=0 show that α+β= ((−b)/a) and αβ= (c/a) hence form an equation whose sum of roots and product of roots are respectively −(1/2) and 2.


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Question Number 33139 by Rio Mike last updated on 11/Apr/18

$α a n d β a r e r o o t s o f$ ${a x}^{2} + b x + c = 0$ $s h o w t h a t α + β = \frac{- b}{a} a n d α β = \frac{c}{a}$ $h e n c e f o r m a n e q u a t i o n w h o s e$ $s u m o f r o o t s a n d p r o d u c t o f r o o t s$ $a r e r e s p e c t i v e l y$ $- \frac{1}{2} a n d 2 .$
	Answered by MJS last updated on 11/Apr/18

	${a x}^{2} + b x + c = 0$ $x^{2} + \frac{b}{a} x + \frac{c}{a} = 0$ $(x - α) (x - β) = 0$ $x^{2} - (α + β) x + α β = 0$ $\Rightarrow α + β = - \frac{b}{a}; α β = \frac{c}{a}$ $x^{2} + \frac{1}{2} x + 2 = 0$ $x = - \frac{1}{4} \pm \sqrt{\frac{1}{16} - 2} = - \frac{1}{4} \pm \frac{\sqrt{31}}{4} i$ $α = - \frac{1}{4} - \frac{\sqrt{31}}{4} i$ $β = - \frac{1}{4} + \frac{\sqrt{31}}{4} i$ $α + β = - \frac{1}{2}$ $α β = (- \frac{1}{4} - \frac{\sqrt{31}}{4} i) (- \frac{1}{4} + \frac{\sqrt{31}}{4} i) =$ $= \frac{1}{16} - \frac{31}{16} i^{2} = \frac{1}{16} + \frac{31}{16} = 2$
	Commented by Rio Mike last updated on 11/Apr/18

	$G e n e r a l l y t h e n e w e q u a t i o n i s$ $2 x^{2} + x + 4 = 0$ $s i n c e$ $S u m o f r o o t s α + β = - \frac{1}{2}$ $α β = 2$ $N e w e q u a t i o n = X^{2} - (S u m r o o t s) X + p r o d u c t o f r o o t s$ $x^{2} - - \frac{1}{2} x + 2$ $2 x^{2} + x + 4 = 0$
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