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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 4 months ago

Easy Math Editor

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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 \(A\).

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Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestWhat 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 \(A\).Log in to reply