Question

Find the number of values of \(n\) between \(1\) and \(100\) such that \(x^2+x-n=0\) has integer roots.

Answer 1

there's only 2?

Answer 2

not sure im still working on this haha! but my first guess would be more than 2

Answer 3

I mean only two for each n with consecutive factor pairs

Answer 4

so I would expect there to be 9 total

Answer 5

pairs

Answer 6

with non distinct values

Answer 7

x^2+x-2 = (x-1)(x+2)

how did u get 9 pairs so fast ! xD

Answer 8

n and n+1, where 1<n<9?

Answer 9

well, they have to be consecutive to give us a positive 1 in the middle for the coefficient right? they cannot multiply to more than 100, so since 10*11 >100, we only need pairs less than 10

Answer 10

Brilliant !!!

Answer 11

(-1,2)=-2
(-2,3)=-6
(-3,4)=-12
yada yda till we hit, (-9,10)=-90

Answer 12

I should've used a diff variable, not n. But same thing.

Answer 13

so the limit on n makes our life easier

Answer 14

I was trying quadratic formula and making it hard

Answer 15

oh, lol. I was about to say, this is an easy one for your genius mind

Answer 16

\[x = \dfrac{-1\pm \sqrt{1+4n}}{2}\]

\(1+4n = m^2\) etc..

Answer 17

Well they have to be integers, so that makes it easy

Answer 18

my start was wrong.. I see it is easy with pairs idea :)

Answer 19

yea, no forest for trees :)

Answer 20

I'm just excited because I actually answered a question you asked!

Answer 21

this is the fastest somebody ever helped me haha xD

Answer 22

lol

Answer 23

The question looks a lot more complex when Openstudy is being weird...

Answer 24

omg! that happened to me yesterday too... i think it is some latex related bug

Answer 25

... that problem would be insane,

Answer 26

just try it with 111 and 100,100,100 for your n.... we would have 12-> like somewhere between10,000 and 11,000 for our restraints... That's alot of pairs

Answer 27

I'm pretty sure @ganeshie8's sole reason for asking this question, was so that he would have equal numbers of fan testimonials as questions asked.

Answer 28

lol

Answer 29

lol that would be just a coincidence XD

Fib the answer would be number of perfect squares between 111 and 100,100,100 right ?

Answer 30

(from quadratic formula)

Answer 31

no,

Answer 32

it would be the number of consecutive pairs within \(12\le x,y\le 10005\)

Answer 33

Ahh okay, i think it should be number of ODD perfect squares between `111` and `1+4*100,100,100`

Answer 34

that should be something like 10005-12=9993

Answer 35

by my method

Answer 36

YES! that kinda checks out with my method too

number of perfect squares less than `1+4*100,100,100` would be `floor(sqrt(1+4*100,100,100))` = 20010

subtract perfect squares less than 111 and divide by 2

Answer 37

I don't know how to find all of those perfect squares though

Answer 38

ikr :) question is about the count, not the values of n :P

Answer 39

true haha, but we did if your method works, just find a way to determine how many perfect squares are between two numbers

Answer 40

number of perfect squares between \(a\) and \(b\) would be somewhere around \(\sqrt{b} - \sqrt{a}\) right ?

Answer 41

assuming \(a < b\)

Answer 42

uhm, that I do not know, shall we attempt that?

Answer 43

see if we get close to my and your answer

Answer 44

idk thats just a ballpark number....

Answer 45

mine should be exact

Answer 46

do we agree number of perfect squares not greater than \(n\) is \(\lfloor \sqrt{n}\rfloor \)

Answer 47

exampel :
number of perfect squares not greater than \(256\) is \(\lfloor \sqrt{256}\rfloor = 16 \) :
\[\{1^2,2^2,3^2,\ldots, 16^2\}\]

Answer 48

9994.46809437 Pretty dang close

Answer 49

I don't know the floor function

Answer 50

floor(2.9) = 2
floor(2) = 2
floor(1.9) = 1

floor(n) is the greatest integer less than or eqal to n

Answer 51

ohk, so where did you get that theorem about perfect squares from?

Answer 52

observation

Answer 53

hmmm, I think we may have just derived the phi function or something like that from number theory, but I'm tired, and thus, cannot be bothered to check if that is the case

Answer 54

thats okay but it has nothing to do with phi function

Answer 55

it's one of those number theory thingies i think