Find all real value of a for which the equation `x^4 + (a - 1)x^3+ x^2 + (a - 1)x + 1 = 0` possesses at least two distinct positive roots

Find all real values of a for which the equation x^4+(a-1)x^3+x^2+(a-1)x+1=0 possesses at least two distinct positive roots.

The real values of 'a' for which the quadratic equation 2x^2 - (a^3 + 8a - 1) x + a^2 - 4a = 0 possess roots of opposite sign is given by:

Find the values of a for which all roots of the equation 3x^(4) + 4x^(3) – 12x^(2) + a = 0 are real and distinct.

Find the values of k for which the equation x^2-4x+k=0 has distinct real roots.

Find the values of k for which the equation x^(2)-4x+k=0 has distinct real roots.

The number of positive integral values of m , m le 16 for which the equation (x^(2) +x+1) ^(2) - (m-3)(x^(2) +x+1) +m=0, has 4 distinct real root is:

The number of positive integral values of m , m le 16 for which the equation (x^(2) +x+1) ^(2) - (m-3)(x^(2) +x+1) +m=0, has 4 distinct real root is:

Find the values of a for which all the roots of the equation x^4-4x^3-8x^2+a=0 are real.