solve-for-x-x-2-10-x-57-4-0-

Question Number 204511 by Ghisom last updated on 20/Feb/24

solve for x

x^{2} - 10 ⌊ x ⌋ + \frac{57}{4} = 0

Answered by mr W last updated on 20/Feb/24

⌊ x ⌋ = n \in Z^{+}

x = n + f w i t h 0 ⩽ f < 1

{(n + f)}^{2} - 10 n + \frac{57}{4} = 0

10 n - \frac{57}{4} = {(n + f)}^{2} ⩾ n^{2}

n^{2} - 10 n + \frac{57}{4} ⩽ 0

\Rightarrow n ⩾ 5 - \sqrt{25 - \frac{57}{4}} \Rightarrow n ⩾ 2

\Rightarrow n ⩽ 5 + \sqrt{25 - \frac{57}{4}} \Rightarrow n ⩽ 8

10 n - \frac{57}{4} = {(n + f)}^{2} < {(n + 1)}^{2}

n^{2} - 8 n + \frac{61}{4} > 0

\Rightarrow n < 4 - \sqrt{16 - \frac{61}{4}} \Rightarrow n ⩽ 3

\Rightarrow n > 4 + \sqrt{16 - \frac{61}{4}} \Rightarrow n ⩾ 5

\Rightarrow n = 2, 3, 5, 6, 7, 8

\Rightarrow x = \sqrt{10 n - \frac{57}{4}}

n = 2 : x^{2} = 20 - \frac{57}{4} \Rightarrow x = \frac{\sqrt{23}}{2} ✓

n = 3 : x^{2} = 30 - \frac{57}{4} \Rightarrow x = \frac{\sqrt{63}}{2} ✓

n = 5 : x^{2} = 50 - \frac{57}{4} \Rightarrow x = \frac{\sqrt{143}}{2} ✓

n = 6 : x^{2} = 60 - \frac{57}{4} \Rightarrow x = \frac{\sqrt{183}}{2} ✓

n = 7 : x^{2} = 70 - \frac{57}{4} \Rightarrow x = \frac{\sqrt{223}}{2} ✓

n = 8 : x^{2} = 80 - \frac{57}{4} \Rightarrow x = \frac{\sqrt{263}}{2} ✓

Commented by Ghisom last updated on 20/Feb/24

thank you