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.

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.

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

Suppose the cubic x^(3)-px+q has three distinct real roots, where pgt0 and qgt0 . Then find the max and min points .

If the equation x^(2)-px+q=0 and x^(2)-ax+b=0 have a comon root and the other root of the second equation is the reciprocal of the other root of the first, then prove that (q-b)^(2)=bq(p-a)^(2) .

Let a ,b , c ,p ,q be real numbers. Suppose alpha,beta are the roots of the equation x^2+2p x+q=0,alphaa n d1//beta are the roots of the equation a x^2+2b x+c=0,w h e r ebeta^2 !in {-1,0,1}dot Statement 1: (p^2-q)(b^2-a c)geq0 Statement 2: b!=p aorc!=q a