Q.Form the quadratic equation whose roots are $\dfrac{p-q}{p+q}$, $-\dfrac{(p+q)}{p-q}$ $(p \neq \pm q)$.

Compute the sum and product of the given roots, then build $x^2-(\text{sum})x+(\text{product})=0$ and clear denominators.

Let $\alpha=\dfrac{p-q}{p+q}$ and $\beta=-\dfrac{p+q}{p-q}$.

Sum:

$\alpha+\beta = \frac{p-q}{p+q}-\frac{p+q}{p-q} = \frac{(p-q)^2-(p+q)^2}{(p+q)(p-q)} = \frac{-4pq}{p^2-q^2}.$

Product:

$\alpha\beta = \frac{p-q}{p+q}\cdot\left(-\frac{p+q}{p-q}\right) = -1.$

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