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Find the product of the three smallest, positive, non-integer solutions to \(\lfloor x \rfloor \lceil x \rceil = x^2.\)

 May 9, 2019
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If   0 < x < 1 ,     \(\lfloor x \rfloor \lceil x \rceil = 0\cdot1=0\)     but     \(0\nless\sqrt0\)

 

If   1 < x < 2 ,     \(\lfloor x \rfloor \lceil x \rceil = 1\cdot2=2\)     and    \(1<\sqrt2<2\)

 

So   \(x=\sqrt2\)   is a solution to  \(\lfloor x \rfloor \lceil x \rceil = x^2\)

 

If   2 < x < 3 ,     \(\lfloor x \rfloor \lceil x \rceil = 2\cdot3=6\)     and    \(2<\sqrt6<3\)

 

So   \(x=\sqrt6\)   is a solution to  \(\lfloor x \rfloor \lceil x \rceil = x^2\)

 

If   3 < x < 4 ,     \(\lfloor x \rfloor \lceil x \rceil = 3\cdot4=12\)     and    \(3<\sqrt{12}<4\)

 

So   \(x=\sqrt{12}\)   is a solution to  \(\lfloor x \rfloor \lceil x \rceil = x^2\)

 

The product of the three smallest, positive, non-integer solutions  =  \(\sqrt2\cdot\sqrt6\cdot\sqrt{12}\,=\,\sqrt{144}\,=\,12\)

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 May 9, 2019

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