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

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In summary, the given problem presents an equation of a^2-b^2+cd^2=2022 and asks for the variables, minimum value, and solving method. The variables in the equation are a, b, c, and d. The minimum value of a+b+c+d cannot be determined without additional information or constraints and can possibly be negative. Different mathematical methods, such as substitution or linear combinations, can be used to solve the problem depending on the given constraints.
  • #1
anemone
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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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  • #2
Thanks for the interesting problem. I guess
[tex]a=1,b=2,c=1,d=45;\ a+b+c+d=49[/tex]
I would like to know the right answer.
 
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  • #3
anuttarasammyak said:
Thanks for the interesting problem. I guess
[tex]a=1,b=2,c=1,d=45;\ a+b+c+d=49[/tex]
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.
 
  • #4
The best I can do for now is:$$a = 17, b = 1, c = 6, d = 17; \ a + b + c + d = 41$$
 
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  • #5
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##
 
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  • #6
Opps! I didn't see @Steve4Physics 's answer above soon enough. Why can't I delete my wrong answer (27)?
 
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FAQ: What is the minimum value of a+b+c+d if a^2-b^2+cd^2=2022?

1. What is the minimum value of a+b+c+d?

The minimum value of a+b+c+d cannot be determined solely from the equation a^2-b^2+cd^2=2022. More information about the values of a, b, c, and d is needed to find the minimum value.

2. How can the minimum value of a+b+c+d be determined?

To find the minimum value of a+b+c+d, we need to solve for the values of a, b, c, and d that satisfy the equation a^2-b^2+cd^2=2022 and minimize the sum of a, b, c, and d. This can be done using mathematical techniques such as substitution, elimination, or graphing.

3. Is there only one possible solution for the minimum value of a+b+c+d?

No, there can be multiple solutions for the minimum value of a+b+c+d depending on the values of a, b, c, and d that satisfy the equation a^2-b^2+cd^2=2022 and minimize the sum of a, b, c, and d.

4. Can the minimum value of a+b+c+d be negative?

Yes, the minimum value of a+b+c+d can be negative if the values of a, b, c, and d that satisfy the equation a^2-b^2+cd^2=2022 also result in a negative sum. This can occur if one or more of the values of a, b, c, and d are negative.

5. What is the significance of the minimum value of a+b+c+d in this equation?

The minimum value of a+b+c+d is the smallest possible sum of the four variables a, b, c, and d that satisfies the equation a^2-b^2+cd^2=2022. It can be used to find the smallest possible value of the expression a+b+c+d or to determine the minimum amount of resources needed to satisfy the equation.

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