Write the number of quadratic equations, with real roots, which d…

Question

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

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Answer

Let a and b be the real roots of the quadratic equation.
we need to find the number of quadratic equation such that they ramain unchanged even if roots are squared.
$a^2=a \text { and } b^2=b$
$\Rightarrow a(a-1)=0 \text { and } b(b-1)=0$
$\Rightarrow a=0 \text { or } a=1 \text { and } b=0 \text { or } b=1$
so we have four pairs of roots $(0,0),(0,1),(1,0),(1,1)$
For $(0,0)$
$(x-0)(x-0)=x^2$
for $(0,1)$
$(x-0)(x-1)=x(x-1)=x^2-1$
For $(1,0)$
$(x-1)(x-0)=(x-1) x=x^2-1$
For $(1,1)$
$(x-1)(x-1)=(x-1)^2=x^2-2 x+1$
So there are $3$ quadratic equations with real roots, which do not change by squaring their roots.

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