Math Question and Answers


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 65227 by Tawa1 last updated on 26/Jul/19

Commented by Tony Lin last updated on 26/Jul/19

$l e t α, β t w o r o o t s o f x^{2} + a x + 1 = 0$ $α β = 1 a n d α, β a r e i n t e g e r s$ $(i r r a t i o n a l r o o t s c o m e i n p a i r i f t h e$ $c o e f f i c i e n t o f p o l y n o m i a l e q u a t i o n$ $a r e r a t i o n a l b u t t w o e q u a t i o n s h a v e$ $o n l y o n e c o m m o n r o o t, s o α & β a r e$ $r a t i o n a l, a n d a s h o u l d b e a n i n t e g e r$ $s o α & β a r e i n t e g e r s \Rightarrow α = 1, β = 1 o r$ $α = - 1, β = - 1)$ $\Rightarrow α + β = \pm 2$ $a m a y b e \pm 2$ $i f a = 2$ $\Rightarrow x^{2} + x + 2 = 0$ $∴ x h a s n o r e a l r o o t$ $\Rightarrow a \neq 2$ $i f a = - 2$ $\Rightarrow x^{2} + x - 2 = 0$ $\Rightarrow (x + 2) (x - 1) = 0$ $\Rightarrow x = - 2 o r x = 1 (i n t e g e r r o o t)$ $\Rightarrow a = - 2$ $\Rightarrow x^{2} - 2 x + 1 = 0$ $\Rightarrow {(x - 1)}^{2} = 0$ $\Rightarrow x = 1$ $∴ x^{2} + b x + c h a s a r o o t 1$ $\Rightarrow 1 + b + c = 0$ $\Rightarrow b + c = - 1$ $\Rightarrow a + b + c = - 3$
Commented by Tawa1 last updated on 26/Jul/19

$Wow God bless you sir, how is α + β = \pm 2 .$ $i understand α β = 1, because α β = \frac{c}{a}$
Commented by Tawa1 last updated on 26/Jul/19

$Or since α β = 1, then α = 1, β = 1$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum