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if x 3 +px+q=0 has two equal roots then show that 27q 2 +4p 3 =0


if x3+px+q=0 has two equal roots then show that 27q2+4p3=0



 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 

Grade:10

2 Answers

AskiitiansExpert Abhinav Batra
25 Points
13 years ago

dear ajinkya

suppose α is the double root of x3+px+q=0

then sum of roots of x3+px+q=0 is given by -1* coefficient of x2

let β be the thir root of the given equation

then 2α+β=coefficien -1*0=0

so β=-2α

Double   sum of roots of x3+px+q=0 is given by coefficient of x

so α2 +αβ+βα=p

But β=-2α

so α2 -4α2=p

-3α2 =p 

-108α6=4p3-----1

Product of roots of given equation is given by -1*constant term

so   α2 *-2α=-q

     2α3=q 

    108α6=27q2---2

Adding eq 1 and eq 2

4p3 + 27q2=0

ALL THE best

AskIITIANsExpert

ABhinav Batra
soumyabrata pal
18 Points
13 years ago

Let  the roots be x,x,y.

then 2x+y=0

x2 +xy+xy=p

or p=x(x+2y)

or p=x(x-4x)

or p=-3x2

or p3=-27x6

also

x4y2 =q2

or q6=4x6

hence 4p3+27q3=0

proved

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