What is the minimum value of a+b+c+d if a^2-b^2+cd^2=2022?

[anemone]

Here is this week's POTW:

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If $a,\,b,\,c$ and $d$ are non-negative integers and $a^2-b^2+cd^2=2022$, find the minimum value of $a+b+c+d$.

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[anuttarasammyak]

Thanks for the interesting problem. I guess
$$a=1,b=2,c=1,d=45;\ a+b+c+d=49$$
I would like to know the right answer.

 

[anemone]

Thanks for the interesting problem. I guess
$$a=1,b=2,c=1,d=45;\ a+b+c+d=49$$
I would like to know the right answer.

I am sorry. Your guess is incorrect.

I will wait a bit longer before posting the answer to this POTW, just in case there are others who would like to try it out.

 

[PeroK]

The best I can do for now is:$$a = 17, b = 1, c = 6, d = 17; \ a + b + c + d = 41$$

 

[Steve4Physics]

Well, I feel a bit guilty answering as it’s been an (extremely) long time since I was at school! However:
$a=0, b=1, c=7, d=17$
$a+b+c+d = 25$

 

[bob012345]

Opps! I didn't see @Steve4Physics 's answer above soon enough. Why can't I delete my wrong answer (27)?