Question

Form a quadratic equation whose roots are $\frac{p}{q}$ and $\frac{q}{p}$

Answer 1

We know that if $m$ and $n$ are the roots of a quadratic equation $a{x}^2+bx+c=0$, then the sum of the roots is $(m+n)$ and the product of the roots is $\left(mn\right)$. And then the quadratic equation becomes $x^2-(m+n)x+mn=0$

Here, it is given that the roots of the quadratic equation are $m=\frac{p}{q}$ and $n=\frac{q}{p}$, therefore,

The sum of the roots is:

$m+n=\frac{p}{q}+\frac{q}{p}=\frac{(p\times p)+(q\times q)}{pq}=\frac{p^2+q^2}{pq}$

And the product of the roots is:

$mn=\frac{p}{q}\times \frac{q}{p}=1$

Therefore, the required quadratic equation is

$x^2-(m+n)x+mn=0\Rightarrow x^2-\left(\frac{p^2+q^2}{pq}\right)x+1=0\Rightarrow pq{x}^2-(p^2+q^2)x+pq=0$

Hence, $pq{x}^2-(p^2+q^2)x+pq=0$ is the quadratic equation whose roots are $\frac{p}{q}$ and $\frac{q}{p}$.