If `x^(2) +ax - 3x-(a+2) = 0` has real and distinct roots, then minimum value of `(a^(2)+1)//(a^(2)+2)` is

For all real values of x, the minimum value of (1-x+x^(2))/(1+x+x^(2)) is

If (a^(2)-14a+13)x^(2)+(a+2)x-2=0 does not have two distinct real roots, then the maximum value of a^(2)-15a is _________.

Let x^2-3x+p=0 has two positive roots aa n db , then minimum value if (4/a+1/b) is,

If the equation sin ^(2) x - k sin x - 3 = 0 has exactly two distinct real roots in [0, pi] , then find the values of k .

If the equation x^4- λx^2+9=0 has four real and distinct roots, then lamda lies in the interval

If a x^2+b x+6=0 does not have distinct real roots, then find the least value of 3a+bdot

If the cubic 2x^3-9x^2+12 x+k=0 has two equal roots then minimum value of |k| is______.

If the cubic 2x^3-9x^2+12 x+k=0 has two equal roots then minimum value of |k| is______.

If x_(1) and x_(2) are the real and distinct roots of ax^(2)+bx+c=0 then prove that lim_(xtox1) (1+sin(ax^(2)+bx+c))^((1)/(x-x_(1)))=e^(a(x_(1)-x_(2))).

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