How to figure out all even integer values of a

hestyllvql 2022-04-23 Answered
How to figure out all even integer values of a quadratic equation?
I know that the answer to the question asked above is infinity, but i have a question like this- (1)f(x)=1, Therefore, i need all even integral values of that f(x). How do I figure this out?
Say f(x)=(x2)+4x60
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Answers (1)

Giancarlo Brooks
Answered 2022-04-24 Author has 11 answers
Step 1
For integer x,
f(x)=(x+2)264
is an even number iff x+2 is even.
Or
x2+4x60x2mod2
and it is obvious that squaring preserves parity.
Step 2
Note that minimum effort would have been to try and observe the easy pattern
60,57,48,39,28,15,0
Step 3
As noticed by, if x is not restricted to be an integer, the possible values are the solutions of the quadratic equations
x2+4x60=2n.
These values are all different, except for n=0 (and x=10).
Similarly, the odd values occur at the roots of
x2+4x60=2n+1.
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