The equation whose roots are $\frac{p-q}{p+q}$ and $-\frac{p+q}{p-q}$ will be

1. Let the roots be $\alpha =\frac{p-q}{p+q}$ and $\beta =-\frac{p+q}{p-q}$.2. Calculate the product of the roots ($\alpha \beta$):
$$\alpha \beta =\left(\frac{p-q}{p+q}\right)\times \left(-\frac{p+q}{p-q}\right)=-1$$
Calculate the sum of the roots ($\alpha +\beta$):
$$\alpha +\beta =\frac{p-q}{p+q}-\frac{p+q}{p-q}=\frac{(p-q{)}^2-(p+q{)}^2}{(p+q)(p-q)}$$
$$\alpha +\beta =\frac{(p^2-2pq+q^2)-(p^2+2pq+q^2)}{p^2-q^2}=\frac{-4pq}{p^2-q^2}$$
A quadratic equation with a known sum and product of roots is given by:
$$x^2-(\alpha +\beta )x+\alpha \beta =0$$
$$x^2-\left(\frac{-4pq}{p^2-q^2}\right)x+(-1)=0$$
$$x^2+\frac{4pq}{p^2-q^2}x-1=0$$
Multiply the entire equation by $(p^2-q^2)$ to clear the fraction:
$$(p^2-q^2)x^2+4pqx-(p^2-q^2)=0$$