If-the-equation-x-2-2-1-2-x-and-x-2-2-1-2-x-have-one-and-only-one-root-in-common-then-is-equal-to-

Question Number 20684 by Tinkutara last updated on 31/Aug/17

I f t h e e q u a t i o n x^{2} + β^{2} = 1 - 2 β x a n d

x^{2} + α^{2} = 1 - 2 α x h a v e o n e a n d o n l y

o n e r o o t i n c o m m o n, t h e n ∣ α - β ∣ i s

e q u a l t o

Answered by $@ty@m last updated on 02/Sep/17

L e t t h e c o m m o n r o o t b e p

\Rightarrow {(p + α)}^{2} = 1 & {(p + β)}^{2} = 1

C a s e - I . p + α = 1, p + β = 1

\Rightarrow α = β

\Rightarrow b o t h e q u a t i o n s a r e s a m e .

C a s e - I I . p + α = 1 & p + β = - 1

\Rightarrow α - β = 2

C a s e - I I I . p + α = - 1 & p + β = 1

\Rightarrow - (α - β) = 2 \Rightarrow (α - β) = - 2

C o m b i n i n g t h e r e s u l t s o f C a s e I I

a n d C a s e - I I I, w e g e t

∣ α - β ∣= 2

Commented by Tinkutara last updated on 02/Sep/17

Thank you very much Sir!