a problem

The number of quadratic equations, which are unchanged by squaring their roots is ?

8 Answers

1
Athenes Analyst ·

Roots can be 0 and 1 only.
So combinations may be (0,0) ; (1,1) ; (0,1).
So probably 3.

Sorry I am not thinking that much

262
Aditya Bhutra ·

let the roots be x,y

thus on sqarring , roots are x2,y2

there are two possibilities ,

x=x2 and y=y2

x=0,1 y=0,1 (3 solns.)

other possibility ,

x=y2 y=x2

or x = x4 and y=y4

x(x3-1)=0 and y(y3-1)=0

x=0,1,ω,ω2 y=0,1,ω,ω2

thus all solns. are ,

(1,1)
(1,0)
(0,0)
(ω,ω2)

Ans:4

1
Athenes Analyst ·

aditya you want to say ω=ω2.
On squaring ω we dont get the same answer.

262
Aditya Bhutra ·

(ω,ω2)

on squarring ,

(ω2,ω)

same roots , thus same equation.

71
Vivek @ Born this Way ·

THanks!

1
Athenes Analyst ·

Thanks Aditya.... Silly me :D

11
Devil ·

A couple of points:
Firstly for a Quadratic to remain unchanged its coefficients must be unchanged...

Σx2=Σx and Πx=Πx2

So that gives (x+y)=2,-1 with xy=±1

So altogether 4 such equations (It is unnecessary to get the solutions)

Secondly I doubt whether the list of solutions that Aditya has given is exhaustive. I think there are a couple of more solutions (provided I haven't screwed up my calculations again)

262
Aditya Bhutra ·

plz disclose the "couple of more solutions" . (because i am not getting any)

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