Question

Find the quadratic equation whose roots are p+q/p and p+q/q. Also find the nature of roots when p=2 and q=3

Answer 1

Therefore the quadratic equation is

$pqx^2-({p^2+q^2+2pq})x+({p^2+q^2+2pq})=0$

The roots are real.

Step-by-step explanation:

Given roots are $\frac{p+q}{p}$ and $\frac{p+q}{q}$

sum of the root =$\frac{p+q}{p}+\frac{p+q}{q}$$=\frac{p^2+q^2+2pq}{pq}$

product of root = $(\frac{p+q}{p})(\frac{p+q}{q})$= $=\frac{p^2+q^2+2pq}{pq}$

so $\frac{-b}{a}$$=\frac{p^2+q^2+2pq}{pq}$ and $\frac{c}{a}$$=\frac{p^2+q^2+2pq}{pq}$

Therefore the quadratic equation is

$ax^2+bx+c=0$

⇔$a(x^2+\frac{b}{a} x+\frac{c}{a} )=0$

⇔$x^2-\frac{p^2+q^2+2pq}{pq}x+\frac{p^2+q^2+2pq}{pq}=0$

⇔$pqx^2-({p^2+q^2+2pq})x+({p^2+q^2+2pq})=0$

For p= 2 and q=3

Then the root of the equation is $\frac{2+3}{2}$ and $\frac{2+3}{3}$   =$\frac{5}{2}$ and $\frac{5}{3}$

The roots are real.