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Math Help - Solve the equation

  1. #1
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    Solve the equation

    \lfloor x \rfloor =x^2(x^2-2) where \lfloor x \rfloor denotes the floor function.
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  2. #2
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    Quote Originally Posted by james_bond View Post
    \lfloor x \rfloor =x^2(x^2-2) where \lfloor x \rfloor denotes the floor function.
    With the floor on the left, the right side must be integer valued which, in turn means x^2 must be integer valued. x is either an integer or the square root of an integer.

    So just try some integers. If x= 0, this is 0= 0(0-2) which is certainly true!
    If x= 1, 1= 1(1-2)= 0-1 which is NOT true. If x= -1, -1= 1(1-2)= -1 which is true!

    If x= \sqrt{2} this is 1= 2(2-1) which is NOT true.

    For x> 1, the right hand side will be negative so there can be no further negative roots.

    If x= \sqrt{3} this is 1= 3(3-1) which is NOT true.

    If x= 2, 2= 4(4-2)= 8 which is NOT true.

    If x= \sqrt{5} this is 2= 5(5-1)= 20 and it should be clear that if you the difference between right and left side just getting larger and larger- the right side is growing as x^4 while the right side is only growing as x.

    The roots to this equation are x= -1 and x= 0.
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  3. #3
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    Unfortunately \sqrt{1+\sqrt{2}} is also a solution
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