Question

In the domain [0, 100], how many solutions are there for the equation \[x^2- \lfloor x \rfloor x = 81.25\]
How to start?

Answer 1

Instead of thinking of it as one function defined on [1,100], maybe think of it as 100 different function defined on the intervals [n,n+1] as n goes from 0 to 99.

Answer 2

That way you can minimize the frustration the floor function brings to the table.

Answer 3

er, i guess the interval should really be [n,n+1)

Answer 4

Then on each interval, the question is really:\[x^2+nx-81.25=0\]where x ranges from [n,n+1). Not sure if this leads to a solution, but it might be a better way of thinking of the problem.

Answer 5

Domain = [0, 100] :S

Answer 6

wait, the function\[x^2+\lfloor x \rfloor x\]is strictly increasing, isnt it?

Answer 7

then it can have at most one solution.

Answer 8

That means it has 0 :'(

Answer 9

strictly increasing on the interval [0,100] anyways.

Answer 10

well, im not 100% sure on the strictly increasing part, im a little fuzzy on what happens when you jump from one unit interval to the other.

Answer 11

no, its strictly increasing for sure.

Answer 12

Sorry!!! Wait... a mistake in the question posted here!!

Answer 13

ah ok, that makes more sense.

Answer 14

I'm sorry!!!! :(

Answer 15

if you look at the function:\[x^2-\lfloor x \rfloor x\]restricted to an interval {n,n+1) (open on the right), note that the values start at 0 when x = n, and as x approaches n+1, the values also approach n+1: