x^3(x+1)=(x+k)(x+2k)

where k belongs to (0.75,1)

find-

i)no. of real roots

ii) greatest real root

iii) least real root

plz help me to solve this or guide me approaching this sum and more such types.......

thanks in advance........

The no of roots of this equation are the no of points of intersection of y=x^4+x^3 and y=x^2+3kx+2k^2.

Differentiating y=x^4+x^3 wrt x we get dy/dx=4x^3+3x^2=x^2(4x+3). dy/dx>0 when x> -3/4, dy/dx

The graph of y=x^2+3kx+2k^2= (x-3k/2)^2-k^2/4 is a parabola with vertex at

(-3k/2,-k^2/4) By plotting these graphs we observe that there are two points of intersections. So the equation has two distinct real roots for each k belongs to (0.75,1) .

The no of roots of this equation are the no of points of intersection of y=x^4+x^3 and y=x^2+3kx+2k^2.
Differentiating y=x^4+x^3 wrt x we get dy/dx=4x^3+3x^2=x^2(4x+3). dy/dx>0 when x> -3/4, dy/dx
The graph of y=x^2+3kx+2k^2= (x-3k/2)^2-k^2/4 is a parabola with vertex at
(-3k/2,-k^2/4) By plotting these graphs we observe that there are two points of intersections. So the equation has two distinct real roots for each k belongs to (0.75,1) .