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

```
7 years ago

25 Points
```										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
ABhinav Batra

```
7 years ago
soumyabrata pal
18 Points
```										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
```
7 years ago
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