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

If the cubic equation 3x^3 +px +5=0 has exactly one real root then show that p gt 0

If x^(2) - px + q = 0 has equal integral roots, then

The equation 2x ^(3) -3x ^(2) +p=0 has three real roots. Then find the minimum value of p.

If the equation x^(2)-2px+q=0 has two equal roots, then the equation (1+y)x^(2)-2(p+y)x+(q+y)=0 will have its roots real and distinct only, when y is negative and p is not unity.

If the roots of the equation x^(3) - px^(2) + qx - r = 0 are in A.P., then prove that, 2p^3 −9pq+27r=0

If one root of the equation x^2+px+q=0 is the square of the other then

If he equation x^3+a x^2+b x+216=0 has three real roots in G.P., then b//a has the value equal to _____.

If he equation x^3+a x^2+b x+216=0 has three real roots in G.P., then b//a has the value equal to _____.

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