Question

Write the number of quadratic equations, with real roots, which do not change by squaring their roots.

Answer 1

Let a and b be the real roots of the quadratic equation.
We need to find the number of quadratic equations such that they remain unchanged even if roots are squared.
$a^2=aand{b}^2=b\Rightarrow a^2-a=0and{b}^2-b=0\Rightarrow a(a-1)=0andb(b-1)=0\Rightarrow a=0ora=1andb=0orb=1\text{So we have four pair of roots}(0,0),(0,1),(1,0),(1,1)For(0,0)(x-0),(x-1)=x(x-1)=x^2-1(x-1)(x-0)=(x-1)x=x^2-1For(1,1)(x-1)(x-1)=(x-1{)}^2=x^2-2x+1$\\

So there are 3 quadratic equations with real roots, which do not change by squaring those roots.