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Floor functions

Find all real solutions to the equation $$4{x}^2 - 40\lfloor{x}\rfloor + 51 = 0$$. Here, if $$x$$ is a real number, then $$\lfloor{x}\rfloor$$ denotes the greatest integer that is less than or equal to $$x$$.

Note by Dhrubajyoti Ghosh
1 month ago

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What have you tried?

Hint: Since $$40 \lfloor x \rfloor$$ and 51 are integers, then so is $$4x^2$$.

Hint 2: Bound it. Express $$x$$ as $$\lfloor x \rfloor + \{ x \}$$. Separate the integer part and the fractional part.

Hint 3: $$x$$ can be expressed as $$\frac{\sqrt A}4$$ for some integer $$A$$.

- 1 month ago