Question

The greatest integer, [x], is equal to the largest integer that is smaller than or equal to x. In the domain [0,100],
how many solutions are there to:
\[x^2 - [x]x = 81.25\]

Answer 1

the answer is D :)

Answer 2

fizi stop trolling -.-

Answer 3

100-81.25

Answer 4

why though batman

Answer 5

k :/

Answer 6

k thanks bye close

Answer 7

dumb it down for me and explain

Answer 8

Well, factoring might help: \[
x(x-\lfloor x\rfloor)=81.25
\]

Answer 9

legit way ^

Answer 10

Suppose \(x = \lfloor x\rfloor +a\). We know \(0\leq a < 1\)

Answer 11

Also:\[
x-\lfloor x\rfloor = a
\]

Answer 12

so \[
xa = 81.25
\]

Answer 13

\[
a = \frac{81.25}{x}\implies 0\leq \frac{81.25}{x}<1
\]

Answer 14

So \[
81.25 < x\leq 100
\]

Answer 15

ah, makes sense, ty.

Answer 16

Sham get off that website, if you're going to cheat.

Answer 17

<3

Answer 18

this is about learning batman shutup

Answer 19

if i was aiming for just an answer i would'nt have asked for an explanation

Answer 20

I was joking

Answer 21

I wasn't sure what I was doing. I winged it.

Answer 22

It's pretty much right lol

Answer 23

worked out in the end xD

Answer 24

Did you check it?

Answer 25

Your answer should be 19 sham, ye?

Answer 26

My solution is not complete.

Answer 27

I only really gave a feasible range, but there will be many gaps in it.

Answer 28

I didn't put restrictions on \(a\) despite the fact that \(a=x-\lfloor x\rfloor\)

Answer 29

:)

Answer 30

if we said \(\lfloor x\rfloor =82\) then:\[
a = \frac{81.25}{82}
\]

Answer 31

yeah, they wanted the integer solution so i put 19 batman.

Answer 32

So \(x =82+ 81.25/82\) is perhaps a solution... the reasoning make sense but it's important to verify.

Answer 33

What? If \(x\) is an integer then \(x^2-x\lfloor x \rfloor = x^2-x^2=0\)

Answer 34

I would say all solutions are: \[
\left\{\lfloor x\rfloor +\frac{81.25}{\lfloor x\rfloor }\bigg|x\in (81.25,100]\right\}
\]

Answer 35

Are you guys talking about a different problem?

Answer 36

Same

Answer 37

Why 19?

Answer 38

One small alteration:\[
\left\{\lfloor x\rfloor +\frac{81.25}{ x }\bigg|x\in (81.25,100]\right\}
\]

Answer 39

x^2-[x]x=81.25
x-[x]=81.25/x

Answer 40

Why 19 though?