If the equation `x^(3) +px +q =0` has three real roots then show that `4p^(3)+ 27q^(2) lt 0`.

Let 'p' & 'q' be real numbers . Prove that the cubic y = x^(3) + p x + q has three distinct real roots, if 4 p^(3) + 27 q^(2) lt 0 .

If all the roots of x^(3)+px+q=0p,q in R,q!=0 are real then

if 2+isqrt3 be a root of the equation x^(2) + px + q =0 , where p and q are real, then find p and q

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 one root of x^(2) + px+12 = 0 is 4, while the equation x ^(2) + px + q = 0 has equal roots, then the value of q is

If p and q are the root of the equation x^(2) - px + q = 0 , then what are the vlaues of p and q respectively ?

If the difference between the roots of the equation x^(2)-px+q=0 is the same as the difference of the roots of the equation x^(2)-qx+p=0 ,show that p+q+4=0 or p=q.

If the equation x^(3)+px^(2)+qx+1=0(p