H S Hall and S R Knight Solutions for Chapter: The Theory of Quadratic Equations, Exercise 3: EXAMPLES. IX. c.

Show that the expression $A\left(x^2-y^2\right)-xy\left(B-C\right)$ always admits of two real linear factors.

If the equations $x^2+px+q=0,x^2+p^{\text{'}}x+q^{\text{'}}=0$ have a common root, show that it must be equal to $\frac{p{q}^{\text{'}}-p^{\text{'}}q}{q-q^{\text{'}}}$ or $\frac{q-q^{\text{'}}}{p^{\text{'}}-p}$

Find the condition that the expressions $l{x}^2+mxy+n{y}^2, l^{\text{'}}{x}^2+m^{\text{'}}xy+n^{\text{'}}{y}^2$ may have a common linear factor.

If the expression $3x^2+2pxy+2y^2+2ax-4y+1$ can be resolved into linear factors, prove that $p$ must be one of the roots of the equation $p^2+4ap+2a^2+6=0$

Find the condition that the expression $a{x}^2+2hxy+b{y}^2, a^{\text{'}}{x}^2+2h^{\text{'}}xy+b^{\text{'}}{y}^2$ may be respectively divisible by factors of the form $y-mx, my+x$

Show that the equation $x^2-3xy+2y^2-2x-3y-35=0$ for every real value of $x,$ there is a real value of $y,$ and for every real value of $y$ there is a real value of $x$.

If $x$ and $y$ are two real quantities connected by the equation $9x^2+2xy+y^2-92x-20y+244=0$ then $x$ will lie between $3$ and $6,$ and $y$ between $1$ and $10.$

If $\left(a{x}^2+bx+c\right)y+a^{\text{'}}{x}^2+b^{\text{'}}x+c^{\text{'}}=0,$ find the condition that $x$ may be a rational function of $y.$