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

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

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