show that the roots of the equation x^2 −2x=(b−c)^2 −1 are rational if b and c are rational numbers.


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Question Number 90772 by MWSuSon last updated on 26/Apr/20

$s h o w t h a t t h e r o o t s o f t h e e q u a t i o n$ $x^{2} - 2 x = {(b - c)}^{2} - 1 a r e r a t i o n a l i f$ $b a n d c a r e r a t i o n a l n u m b e r s .$
Commented by jagoll last updated on 26/Apr/20

$x^{2} - 2 x + 1 = {(b - c)}^{2}$ ${(x - 1)}^{2} - {(b - c)}^{2} = 0$ $(x - 1 + b - c) (x - 1 + c - b) = 0$ $x_{1} = c + 1 - b & x_{2} = b + 1 - c$ $i f b, c r a t i o n a l n u m b e r t h e n$ $x_{1} \land x_{2} r a t i o n a l n u m b e r$
Commented by MWSuSon last updated on 26/Apr/20

$t h a n k y o u s i r, i a p p r e c i a t e$
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