 # 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- ${\left(-1\right)}^{f}\left(x\right)=1$, Therefore, i need all even integral values of that f(x). How do I figure this out?
Say $f\left(x\right)=\left({x}^{2}\right)+4x-60$
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Step 1
For integer x,
$f\left(x\right)={\left(x+2\right)}^{2}-64$
is an even number iff $x+2$ is even.
Or
${x}^{2}+4x-60\equiv {x}^{2}\text{mod}2$
and it is obvious that squaring preserves parity.
Step 2
Note that minimum effort would have been to try and observe the easy pattern
$-60,-57,-48,-39,-28,-15,0\cdots$
Step 3
As noticed by, if x is not restricted to be an integer, the possible values are the solutions of the quadratic equations
${x}^{2}+4x-60=2n.$
These values are all different, except for $n=0$ (and $x=-10$).
Similarly, the odd values occur at the roots of
${x}^{2}+4x-60=2n+1.$