Question

short fun question :D

Answer 1

\[[x^2]+[2x]=3x\]where [ . ] -->greatest integer function

Find the values of x which satisfy this equation \[x \in [0,6]\]

Answer 2

0,1,2.666 ?

Answer 3

2.666 is wrong :)
please tell ur method too (:

Answer 4

oh ya... just 0,1 then?

Answer 5

ehh.......its the probably the most bone headed way to approach it lol :)

Answer 6

RHS has to lie between 0 and 18. so i can substitute integer values for LHS which give me less than 18. so thats 0,1,2,3.

if the RHS has to be an integer then, x=0,0.333,0.666.....3.

sooo..
ya...i actually tried all of them

Answer 7

(: ok here is my approach-

the greatest integer function will always give out an integer :D

so
[x^2] -->an integer
[2x]---->an integer
[x^2]+[2x]--->integer+integer =integer :D

so 3x has to be an integer so x has to be integer
so x can be either 0,1,2,3,4,5,6
so because x is an integer we can write
\[[x^2]+[2x]=x^2+2x\]
so now we are left with a quadratic
\[x^2+2x=3x \]solving it we get x=1,0 :)

Answer 8

but x need not be an integer, i.e 0.3333*3=1 (ish)

Answer 9

we take exact values B)

Answer 10

do u agree that the LHS will give an integer?

Answer 11

yep

Answer 12

and LHS=RHS so RHS will also be an integer?

Answer 13

not necessarily,...if that was the case then mentioning greatest integer function would have been pointless.

Answer 14

greatest integer function was just to confuse (:

Answer 15

ok :)

Answer 16

@imqwerty
aagh...i think 2.666 would work. greatest integer function rounds down so [2.66]=2
therefore LHS=8 and 3*2.666=8 (almost).

i guess the answer to this is matter of opinion aswell XD

Answer 17

XD but it can never be an integer haha :D it is almost close tho ahaha

Answer 18

\(x=5/3\) is also a solution
http://www.wolframalpha.com/input/?i=solve+floor%28x%5E2%29%2Bfloor%282x%29%3D3x%2C+0%3C%3Dx%3C%3D6

Answer 19

5/3 =1.666
[5/3]=1 so LHS=3 RHS=5

Answer 20

x=2.6666=8/3, but would not work.
[x^2]+[2x]
=[(8/3)^2]+[2*8/3]
=[64/9]+[16/3]
=7+5
=12

3x=3(8/3)=8 \(\ne12\)
So x=8/3 does not work.

But 5/3 works!

Answer 21

hmmm..so while doing such questions we must also search for rational solutions with the denominator which gets cancelled ... :)