Solve equation 2⌊x⌋ = (x^( 2) /2) (x ∈ R )


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Question Number 173572 by mnjuly1970 last updated on 13/Jul/22

$S o l v e e q u a t i o n$ $2 ⌊ x ⌋ = \frac{x^{2}}{2} (x \in R)$
	Answered by MJS_new last updated on 13/Jul/22
	$x=i+f, i∈Z∧0≤f<1 4i=(i+f)^2 f=−i±2(√i) ⇒ i≥0 0≤f<1 ⇒ f=−i+2(√i)∧i≤4∧i≠1 ⇒ i∈{0, 2, 3, 4} ⇒ x∈{0, 2(√2), 2(√3), 4}$
	$x = i + f, i \in Z \land 0 ⩽ f < 1$ $4 i = {(i + f)}^{2}$ $f = - i \pm 2 \sqrt{i}$ $\Rightarrow i ⩾ 0$ $0 ⩽ f < 1 \Rightarrow f = - i + 2 \sqrt{i} \land i ⩽ 4 \land i \neq 1$ $\Rightarrow i \in {0, 2, 3, 4}$ $\Rightarrow x \in {0, 2 \sqrt{2}, 2 \sqrt{3}, 4}$
	Commented by mnjuly1970 last updated on 14/Jul/22

	$t h a n k y o u s o m u c h . .$
	Answered by mr W last updated on 13/Jul/22

	$n = ⌊ x ⌋ ⩾ 0 \Rightarrow x ⩾ 0$ $4 n = {(n + f)}^{2} ⩾ n^{2} \Rightarrow n ⩽ 4$ $4 n = {(n + f)}^{2} < {(n + 1)}^{2} \Rightarrow n \neq 1$ $\Rightarrow n = 0, 2, 3, 4$ $\Rightarrow x = 2 \sqrt{n} = 0; 2 \sqrt{2}; 2 \sqrt{3}; 4$
	Commented by mnjuly1970 last updated on 14/Jul/22

	$g r a t e f u l s i r$
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