show that the quadratic equation xpxqk2 has two distinct rea

show that the quadratic equation (x-p)(x-q)=k^2 has two distinct real roots p cannot equal q

Solution

A Quadratic equation has real distinct roots if discriminant >0

Here x^2-px-qx+pq=k^2

x^2-(p+q)x +pq-k^2=0

-(p+q)^2-4(pq-k^2)>0

p^2 +q^2-2pq -4k^2>0

(p-q)^2-4k^2>0

as p is not equal to q so (p-q)^2 is not equal to 0.

k^2 is positive quantity

so equation has two distinct real roots for (p-q)^2>k^2.