Problema Solution
If the sum of the roots of the quadratic equation 1/x+p + 1/x+q = 1/r is zero then show that the product of their roots is -(p^2+q^2)/2
Answer provided by our tutors
1/x+p + 1/x+q = 1/r.
(x+p+x+q)/(x+p)(x+q)=1/r.
(2x+p+q)/(x^2 +(p+q)x+pq)=1/r.
x^2 +(p+q)x+pq=2xr+r(p+q).
x^2+(p+q-2r)+(pq-pr-qr).
Sum of roots =-b/2a =0 b=0.
p+q-2r=0.
p+q=2r.
prodcut of roots =c/a.
=(pq-pr-qr)
=(pq-(p+q)*r).
Now susbtitute p+q=2r in above equation .
=pq-(p+q)*(p+q)/2.
=(2pq-(p+q)^2)/2.
=(2pq-p^2-q^2-2pq)/2.
=-(p^2+q^2)/2.
Thus proved.