H S Hall and S R Knight Solutions for Chapter: The Theory of Quadratic Equations, Exercise 1: EXAMPLES. IX. a.

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

Prove that the roots of $\left(x-a\right)\left(x-b\right)=h^2$ are always real.

If $x_1$ and $x_2$ are the roots of $a{x}^2+bx+c=0$, then find the value of ${\left(a{x}_1+b\right)}^{-2}+{\left(a{x}_2+b\right)}^{-2}$.

If $x_1$ and $x_2$ are the roots of $a{x}^2+bx+c=0,$ then find the value of ${\left(a{x}_1+b\right)}^{-3}+{\left(a{x}_2+b\right)}^{-3}$.

Find the condition that the one root of $a{x}^2+bx+c=0$ shall be $n$ times of the other.

If $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$, form the equation whose roots are ${\alpha }^2+{\beta }^2$ and ${\alpha }^{-2}+{\beta }^{-2}$.

Form the equation whose roots are square of the sum and of the difference of the roots of $2x^2+2 (m+n) x+m^2+n^2=0$.

Discuss the sign of the roots of the equation

$p{x}^2+qx+r=0$