SOLUTION: Let A and B be the roots of an equation x2 +ax+b=0 and let C and D be the roots of x2+cx+d=0. Express (A-C) (B-C) (A-D) (B-D) in terms of the coefficients a, b,c ,d.

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Question 1077969: Let A and B be the roots of an equation x2 +ax+b=0 and let C and D be the roots of x2+cx+d=0. Express (A-C) (B-C) (A-D) (B-D) in terms of the coefficients a, b,c ,d.
Answer by ikleyn(52835)   (Show Source): You can put this solution on YOUR website!
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Let A and B be the roots of an equation x2 +ax+b=0 and let C and D be the roots of x2+cx+d=0.
Express (A-C) (B-C) (A-D) (B-D) in terms of the coefficients a, b,c ,d.
~~~~~~~~~~~~~~~~~~~~~~ 

1.  (A-C)*(B-C) =  =  = .

    (I replaced here A+B by -a and replaced AB by b based on Vieta's formulas).



2.  (A-D)*(B-D) =  =  = .

    (Again, I replaced here A+B by -a and replaced AB by b based on Vieta's formulas).



3.  Now  (A-C)*(B-C)*(A-D)*(B-D) =  = 

    =  = (regroup)

    =  = (simplify)

    =  = 

      (replace D+C = -c,   =  = (-c)^2 -2d,  CD = d,  D+C = -c,  (CD)^2 = d^2  based on Vieta's formulas. You will get) 

    =  = (simplify)

    =  = (I just got the expression via coefficients a, b, c and d !) = 

    =  = .

Answer. (A-C)*(B-C)*(A-D)*(B-D) = .


Lesson to learn from this solution: 

     This assignment is for those who firmly knows Vieta's formulas.



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