If p,q,r are in G.P. and the equations, `px^(2) + 2qx + r = 0 and dx^2+2ex + f = 0 ` have a common root, then show that `d/p , e/q, f/r` are in A.P.

If a, b, c are in G.P. and the equation ax^(2) + 2bx + c = 0 and dx^(2) + 2ex + f = 0 have a common root, then show that (d)/(a), (e)/(b), (f)/(c ) are in A.P.

If p, q, r are in G.P and the equation px^2+2qx+r=0 and dx^2+2ex+f=0 have a common root, then show that d/p, e/q, f/r are in AP.

If a,b,c are in G.P., then the equations ax^2+2bx+c =0 and dx^2+2ex+f =0 have a common root if d/a,e/b,f/c are in

a,b,c are positive real numbers forming a G.P. If ax^2+2bx+c=0 and dx^ 2 +2ex+f=0 have a common root, then prove that d/a,e/b,f/c are in A.P.

Show that the equations px^2+qx+r=0 and qx^2+px+r=0 will have a common root if p+q+r=0 or, p=q=r.

Prove that if the equations x^2+px+r=0 and x^2+rx+p=0 have a common root then either p=r or , 1+p+r=0.

If x^2+p x+q=0a n dx^2+q x+p=0,(p!=q) have a common roots, show that 1+p+q=0 .

Prove that, if the equations x^2+px+qr=0 and x^2+qx+pr=0[pneq,rne0] have a common root , then p+q+r=0.

If the roots of the quadratics x^2 - qx +p = 0 and x^2 - px + q = 0 (p!= q) differ by a constant, show.that p + q + 4 = 0.

If , p , q are the roots of the equation x^(2)+px+q=0 , then