If `p, q, r, s in R`, then the equation `(x^(2)+px+3q)(-x^(2)+rx+q)(-x^(2)+sx-2q)=0` has

If p, q, r, s in R , then equaton (x^2 + px + 3q) (-x^2 + rx + q) (-x^2 + sx-2q) = 0 has

If p, q, r are rational then show that x^(2)-2px+p^(2)-q^(2)+2qr-r^(2)=0

If each pair of the three equations x^(2) - p_(1)x + q_(1) =0, x^(2) -p_(2)c + q_(2)=0, x^(2)-p_(3)x + q_(3)=0 have common root, prove that, p_(1)^(2)+ p_(2)^(2) + p_(3)^(2) + 4(q_(1)+q_(2)+q_(3)) =2(p_(2)p_(3) + p_(3)p_(1) + p_(1)p_(2))

For p, q in R , the roots of (p^(2) + 2)x^(2) + 2x(p +q) - 2 = 0 are

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

If p, q, r each are positive rational number such tlaht p gt q gt r and the quadratic equation (p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0 has a root in (-1 , 0) then which of the following statement hold good? (A) (r + p)/(q) lt 2 (B) Both roots of given quadratic are rational (C) The equation px^(2) + 2qx + r = 0 has real and distinct roots (D) The equation px^(2) + 2qx + r = 0 has no real roots

If the roots of the equation (1-q+p^(2)/2)x^(2) + p(1+q)x+q(q-1) +p^(2)/2=0 are equal, then show that p^(2) = 4q

Show that if p,q,r and s are real numbers and pr=2(q+s) , then atleast one of the equations x^(2)+px+q=0 and x^(2)+rx+s=0 has real roots.

If -4 is a root of the equation x^(2)+px-4=0 and the equation x^(2)+px+q=0 has equal roots, then the value of q is

If alpha and beta are the ral roots of x ^(2) + px +q =0 and alpha ^(4), beta ^(4) are the roots of x ^(2) - rx+s =0. Then the equation x ^(2) -4qx+2q ^(2)-r =0 has always (alpha ne beta, p ne 0 , p,q, r, s in R) :