## imqwerty one year ago short fun question :D

1. imqwerty

$[x^2]+[2x]=3x$where [ . ] -->greatest integer function Find the values of x which satisfy this equation $x \in [0,6]$

2. baru

0,1,2.666 ?

3. imqwerty

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

4. baru

oh ya... just 0,1 then?

5. baru

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

6. baru

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

7. imqwerty

(: 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 :)

8. baru

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

9. imqwerty

we take exact values B)

10. imqwerty

do u agree that the LHS will give an integer?

11. baru

yep

12. imqwerty

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

13. baru

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

14. imqwerty

greatest integer function was just to confuse (:

15. baru

ok :)

16. baru

@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

17. imqwerty

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

18. ganeshie8

$$x=5/3$$ 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

19. baru

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

20. mathmate

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!

21. imqwerty

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