Find the value of `k` if `x^3-12x+k=0` has three real distinct roots.

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

Find the minimum integral value of k for which the equation e^(x)=kx^(2) has exactly three real distinct solutions.

Let f(x)=x + 2|x+1|+2| x-1| . Find the values of k if f(x)=k (i) has exactly one real solution, (ii) has two negative solutions, (iii) has two solutions of opposite sign.

The number of values of k for which the equation x^3-3x+k=0 has two distinct roots lying in the interval (0,1) is (a) three (b) two (c) infinitely many (d) zero

If a continuous function f defined on the real line R assume positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative, then the equation f(x)=0 has a root in R. Consider f(x)= ke^(x)-x , for all real x where k is a real constant. For k > 0, the set of all values of k for which y=ke^(x)-x=0 has two distinct roots is

Find the values of k for which the roots are real and equal in the following equations : kx² + 4x + 1 = 0

Find the values of k for which the roots are real and equal in the following equations : 5x^2-4x+2+k(4x^2-2x-1) = 0

Find the value of k for which the given value is a solution of the given equation x^2+3ax+k=0 at x=-a

Find the value of k for which k+12, k-6, 3 are in GP.