Prove that for all real values of p the equation p(x-1)(x-2) + (2x-3) = 0 has real roots.

The given equation is, px-1x-2+2x-3=0px2-3x+2+2x-3=0px2-3p-2x+2p-3=0Now comparing it with ax2+bx+c=0 we get, a=p, b =-3p-2 , c=2p-3D=b2-4ac=3p-22-4×p×2p-3=9p2+4-12p-8p2+12p=p2+4Now since p20 and hence p2+4>0As we can see that D is always positive so roots of px-1x-2+2x-3=0will always be real and distinct.

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