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

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 2a+3b+6c = 0, then show that the equation a x^2 + bx + c = 0 has atleast one real root between 0 to 1.

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.

Statement -1 If the equation (4p-3)x^(2)+(4q-3)x+r=0 is satisfied by x=a,x=b nad x=c (where a,b,c are distinct) then p=q=3/4 and r=0 Statement -2 If the quadratic equation ax^(2)+bx+c=0 has three distinct roots, then a, b and c are must be zero.

Show that the equation x^(3)+2x^(2)+x+5=0 has only one real root, such that [alpha]=-3 , where [x] denotes the integral point of x

Find the value of a if x^3-3x+a=0 has three distinct real roots.

Determine k so that the equation x^(2)-3kx+64=0 has no real roots.

If the equation x^(4)+px^(3)+qx^(2)+rx+5=0 has four positive real roots, find the maximum value of pr .

Let a, b, c, p, q be the real numbers. Suppose alpha,beta are the roots of the equation x^2+2px+ q=0 . and alpha,1/beta are the roots of the equation ax^2+2 bx+ c=0 , where beta !in {-1,0,1} . Statement I (p^2-q) (b^2-ac)>=0 Statement 2 b != pa or c != qa .

If p(q-r)x^2+q(r-p)x+r(p-q)=0 has equal roots, then;