Question

If $\alpha ,\beta$ are the roots of $x^2+px+q=0$, form the equation whose roots are ${(\alpha -\beta )}^2$ and ${(\alpha +\beta )}^2$.

Answer 1

Given that $\alpha ,\beta$ are the roots of $x^2+px+q=0$

We have $\alpha +\beta =-p$ and $\alpha \beta =q$

Now let us consider $x^2+ax+b=0$ be the equation whose roots are $(\alpha -\beta {)}^2,(\alpha +\beta {)}^2$

Sum of roots is $-a=(\alpha -\beta {)}^2+(\alpha +\beta {)}^2=2({\alpha }^2+{\beta }^2)=2(p^2-2q)$

$\Rightarrow a=2(2q-p^2)$

Product of roots is $b=(\alpha +\beta {)}^2(\alpha -\beta {)}^2=\left(p^2\right)(p^2-4q)$

So the required equation is $x^2+2(2q-p^2)x+p^2(p^2-4q)=0$