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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