Show that the expression (ax-b)*(cx-d)/(bx-a)*(dx-c) can have all the values for real x if (a*a)-(b*b) and (c*c)-(d*d) have the same sign.

Hello student,
Please find the answer to your question below
Given(ax-b)*(cx-d)/(bx-a)*(dx-c)
=(acx²-(ad+bc)x+bd)/bdx²-(bc+ad)x+ac
=x²-((d/c)+(b/a))x+(bd/ac)/x²-((c/d)+(a/b))x+(ac/bd)
So by finding the discriminant of the equation from the above we can observe that
the expression (ax-b)*(cx-d)/(bx-a)*(dx-c) can have all the values for real x if (a*a)-(b*b) and (c*c)-(d*d) have the same sign.