If `p , q , ra n ds` are real numbers such that `p r=2(q+s),` then show that at least one of the equations `x^2+p x+q=0` and `x^2+r x+s=0` has real roots.

If p,q,r,s are real and prgt4(q+s) then show that at least one of the equations x^2+px+q=0 and x^2+rx+s=0 has real roots.

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 p , qr are real and p!=q , then show that the roots of the equation (p-q)x^2+5(p+q)x-2(p-q=0 are real and unequal.

If p ,q are real p!=q , then show that the roots of the equation (p-q)x^2+5(p+q)x-2(p-q)=0 are real and unequal.

if p,q,r are real numbers satisfying the condition p + q +r =0 , then the roots of the quadratic equation 3px^(2) + 5qx + 7r =0 are

If a ,b ,p ,q are non zero real numbers, then how many comman roots would two equations: 2a^2x^2-2a b x+b^2=0 and p^2x^2+2p q x+q^2=0 have?

If p ,q , in {1,2,3,4}, then find the number of equations of the form p x^2+q x+1=0 having real roots.

If p ,q , in {1,2,3,4}, then find the number of equations of the form p x^2+q x+1=0 having real roots.

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 p ,q in {1,2,3,4,5} , then find the number of equations of form p^2x^2+q^2x+1=0 having real roots.