Menu Close

solve-for-all-x-x-2-x-2-100-




Question Number 174960 by alcohol last updated on 14/Aug/22
solve for all x   ⌊x^2 ⌋ − ⌊x⌋^2  = 100
$${solve}\:{for}\:{all}\:{x}\: \\ $$$$\lfloor{x}^{\mathrm{2}} \rfloor\:−\:\lfloor{x}\rfloor^{\mathrm{2}} \:=\:\mathrm{100} \\ $$
Answered by mr W last updated on 15/Aug/22
say x=n+f with 0≤f<1  ⌊n^2 +2nf+f^2 ⌋−n^2 =100  ⌊2nf+f^2 ⌋=100  100=⌊2nf+f^2 ⌋<2n+1  ⇒n>49.5 ⇒n≥50  ⌊x^2 ⌋=100+n^2   100+n^2 ≤x^2 <101+n^2   ⇒(√(100+n^2 ))≤x<(√(101+n^2 )) with n≥50
$${say}\:{x}={n}+{f}\:{with}\:\mathrm{0}\leqslant{f}<\mathrm{1} \\ $$$$\lfloor{n}^{\mathrm{2}} +\mathrm{2}{nf}+{f}^{\mathrm{2}} \rfloor−{n}^{\mathrm{2}} =\mathrm{100} \\ $$$$\lfloor\mathrm{2}{nf}+{f}^{\mathrm{2}} \rfloor=\mathrm{100} \\ $$$$\mathrm{100}=\lfloor\mathrm{2}{nf}+{f}^{\mathrm{2}} \rfloor<\mathrm{2}{n}+\mathrm{1} \\ $$$$\Rightarrow{n}>\mathrm{49}.\mathrm{5}\:\Rightarrow{n}\geqslant\mathrm{50} \\ $$$$\lfloor{x}^{\mathrm{2}} \rfloor=\mathrm{100}+{n}^{\mathrm{2}} \\ $$$$\mathrm{100}+{n}^{\mathrm{2}} \leqslant{x}^{\mathrm{2}} <\mathrm{101}+{n}^{\mathrm{2}} \\ $$$$\Rightarrow\sqrt{\mathrm{100}+{n}^{\mathrm{2}} }\leqslant{x}<\sqrt{\mathrm{101}+{n}^{\mathrm{2}} }\:{with}\:{n}\geqslant\mathrm{50} \\ $$
Commented by alcohol last updated on 15/Aug/22
thank you  please can you explain?
$${thank}\:{you} \\ $$$${please}\:{can}\:{you}\:{explain}? \\ $$
Commented by mr W last updated on 15/Aug/22
i thought i have given all necessary  steps.  where do you need more explanation?
$${i}\:{thought}\:{i}\:{have}\:{given}\:{all}\:{necessary} \\ $$$${steps}. \\ $$$${where}\:{do}\:{you}\:{need}\:{more}\:{explanation}? \\ $$
Commented by alcohol last updated on 15/Aug/22
i have no idea about it. if you can propose me a book   to read i will be grateful
$${i}\:{have}\:{no}\:{idea}\:{about}\:{it}.\:{if}\:{you}\:{can}\:{propose}\:{me}\:{a}\:{book}\: \\ $$$${to}\:{read}\:{i}\:{will}\:{be}\:{grateful} \\ $$
Commented by mr W last updated on 15/Aug/22
since you asked this question, i assumed  that you know what the greatest integer  function ⌊x⌋ is. i don′t know any books  about such things.
$${since}\:{you}\:{asked}\:{this}\:{question},\:{i}\:{assumed} \\ $$$${that}\:{you}\:{know}\:{what}\:{the}\:{greatest}\:{integer} \\ $$$${function}\:\lfloor{x}\rfloor\:{is}.\:{i}\:{don}'{t}\:{know}\:{any}\:{books} \\ $$$${about}\:{such}\:{things}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *