The real numbers `x_1, x_2, x_3` satisfying the equation `x^3-x^2+b x+gamma=0` are in A.P. Find the intervals in which `beta and gamma` lie.

Let x_1, x_2, x_3 satisfying the equation x^3 - x^2 + betax + gamma = 0 are in GP where (x_1, x_2, x_3 > 0) , then the maximum value of [beta] + [gamma] + 2 is, [.] is greatest integer function.

Find the number or real values of x satisfying the equation 9^(2log_(9)x)+4x+3=0 .

The roots alpha, beta and gamma of an equation x^(3) - 3 a x^(2) + 3 bx - c = 0 are in H.P. Then,

If alpha , beta , gamma are the roots of the equation x^3 -6x^2 +11 x +6=0 then sum alpha^2 beta =

If alpha, beta and gamma the roots of the equation x^(3) + 3x^(2) - 4x - 2 = 0. then find the values of the following expressions: (i) alpha ^(2) + beta^(2) + gamma^(2) (ii) alpha ^(3) + beta^(3) + gamma^(3) (iii) (1)/(alpha)+(1)/(beta)+(1)/(gamma)

If alpha,beta,gamma are the roots of 4x^3-6x^2+7x+3=0 then find the value of alpha beta +beta gamma+gamma alpha .

Find the number of real solution of the equation (cos x)^(5)+(sin x)^(3)=1 in the interval [0, 2pi]

If alpha, beta and gamma are the roots of the equation x^(3) + 3x^(2) - 24x + 1 = 0 then find the value of (3sqrt(alpha)+3sqrt(beta)+ 3sqrt(gamma)).

If alpha , beta , gamma are the roots of the equation x^3 +px^2 + qx +r=0 then the coefficient of x in cubic equation whose roots are alpha ( beta + gamma ) , beta ( gamma + alpha) and gamma ( alpha + beta) is

If alpha, beta, gamma are the roots of the equation x^(3) + x + 1 = 0 , then the value of alpha^(3) + beta^(3) + gamma^(3) , is