Question

for which integer values of "x"
x^2+19x+92
is a perfect square.
I want to know the procedure

Answer 1

\(ax^2+bx+c\) will become a perfect square if it has same (equal) roots,
like example : if the roots were 2,2 , then it'd be a perfect square..

Answer 2

and for \(ax^2+bx+c \) to be a perfect square,
its discriminant must be 0
you know what a discriminant of a quadratic expression is ?

Answer 3

yes I know about d discriminant but how can we say that roots have to be equal for it to be a perfect square?

Answer 4

i must have mis-read the question....let me think

Answer 5

ok.

Answer 6

do you know how to complete the square ?

Answer 7

yes.

Answer 8

If the roots are equal then you end up with something like (x-a)(x-a) after factorizing your equation, where 'a' is the root.

Answer 9

So that is why you must have repeated roots

Answer 10

can you complete the square for that quadratic expression for me ?

Answer 11

yes.give me a min.

Answer 12

[x+(19/2)]^2 + 3/2

Answer 13

start with
x^2+19x+92=n^2

Answer 14

I tried that but I am not getting what to do with it. @BSwan

Answer 15

its number theory :)

Answer 16

why gave medal :-\
i still dint solve it yet

Answer 17

do you what formula the perfect square should have ?

Answer 18

I gave u the medal coz I got the solution thru ur approach.

Answer 19

could you plz share ur sol ?
cuz i found one too

Answer 20

http://math.stackexchange.com/questions/487363/how-can-i-find-integer-values-for-which-a-given-expression-gives-a-perfect-squar
i got it from hartnn

Answer 21

my hint is :-
the square of any integer is either of the form 3k, or 3k+11

Answer 22

ok lol fair enough :)

Answer 23

tell me the reason behind ur hint,i.e.its logic @BSwan

Answer 24

its using the division algorithm

Answer 25

plz elaborate

Answer 26

so let x=3n,3n+1,3n+2
x^2+19x+92
case 1 :-
(3n)^2+19(3n)+92
9n^2+57n+92=3(3n^2+19n)+2 (mod 3) of the form 3k+2

something like this , but i cant remember xD
so ignore it

Answer 27

ok.