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. $a^2=a$ and $b^2=b \Rightarrow a(a-1)=0$ and $b(b-1)=0 \Rightarrow a=0$ or $a=1$ and $b=0$ 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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