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Problem in finding quad. eqn. from the roots

  1. Aug 13, 2011 #1
    In general, if α(alpha) and ß (beta) are roots of eqn. ax^2 +bx +c=0
    then for finding the equation whose roots are α+2 and ß+2 can be done by
    addition of roots (α+2+ß+2=-b/a) and product of roots (α+2)(ß+2)=c/a
    By solving this we get ax^2 -(4a-b)x + (4a-2b+c)=0

    The problem is this that,by replacing x in place of (x-2)in the given equation
    ax^2 +bx +c=0 we get the same answer ax^2 -(4a-b)x + (4a-2b+c)=0
    but this method ( replacing x by (x-2) ..) is not mentioned anywhere
    i am not able to understand this method
    please help .
     
  2. jcsd
  3. Aug 14, 2011 #2

    HallsofIvy

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    I'm not sure what you are asking. Certainly, it is true that if [itex]\alpha[/itex] and [itex]\beta[/itex] are roots of a quadratic equation, then the equation is of the form [itex]a(x- \alpha)(x- \beta)= 0[/itex] for some number a.

    Similarly, if the roots of a quadratic equation are [itex]\alpha+ 2[/itex] and [itex]\beta+ 2[/itex], then the equation is of the form [itex]a(x- (\alpha+ 2))(x- (\beta+ 2)= 0[/itex] which is the same as [itex]a((x- 2)- \alpha)(x- 2)- \beta)= 0[/itex], the original equation with "x" replaced by "x- 2".
     
  4. Aug 14, 2011 #3
    thanks i got it:smile:
     
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