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Question

The number of quadratic equations (consider leading coefficient as 1) with real roots which remain unchanged when their roots are squared, is

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Solution

Let the two roots be α,β.
f(x)=(xα)(xβ)=0
Roots are squared α2,β2
Roots remain same as even after squaring,
α=α2, β=β2α=0,1β=0,1
(roots are real)

So, α=0,β=0x2=0
α=1,β=1(x1)2=0α=0,β=1x2x=0α=1,β=0x2x=0

Hence, there are 3 quadratic equations possible.

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