Question

the number of quadratic equation having real root and which do not change by squaring their roots is

Answer 1

ANSWER

α2+β2=α+β   ...(1)

α2β2=αβ    ...(2)

⇒αβ(αβ−1)=0

⇒α=0 or β=0 or αβ=1

For α=0

Substituting this in (1)

β2−β=0

⇒β=0,1

Hence roots are 0,0 and 0,1

Similarly for β=0

We get roots are 0,0 and 0,1

And for αβ=1

From (1)

(α+β)2−2αβ=α+β

substitute t=α+β

t2−t−2=0⇒(t−2)(t+1)=0

⇒α+β=2,−1

From α+β=2 and αβ=1

We get root 1,1

And from α+β=−1 and αβ=1

We get root ω,ω²