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Question
If the equations x2 + px + q = 0 and x2 + p’x + q’ = 0 have a common root, show that it must be equal to `("pq'" - "p'q")/("q" - "q")` or `("q" - "q'")/("p'" - "P")`
Solution
If α is the common root, then.
α2 + pα + q = 0 .......(1)
α2 + p’α + q’ = 0 .......(2)
Subtracting α(p – p’) = q’ – q
α = `("q'" - "q")/("p" - "P'") =("q" - "q'")/("p'" - "P")` .......(3)
Eliminating α from (1) and (2)
`"p'"alpha^2 + "pp'"alpha + "p'q" = 0`
`"p'"alpha^2 + "pp'" + "pq" = 0`
(–) (–) (–)
`alpha^2 ("p'" - "p") + "p'q" - "p'q" = 0`
`alpha^2("p'" - "p") = "pq'" - "p'q")`
`alpha^2 = ("pq'" - "p'q")/("p'" - "p")`
`((4))/((3)) => alpha^2/alpha`
= `("pq'" - "p'q")/(("p'" - "p")) xx ("p'" - "p")/("q" - "q'")`
`alpha = ("pq'" - "qp'")/("q" - "q'")`
or
`("q" - "q'")/("p'" - "p")`
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