MHB What is x when c is at maximum?

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The discussion revolves around solving the equations a = x^2 + 2cx + 1 and b = 2x + 3c + 3, with the condition that a/b = 5 plus a remainder of 11. The solution for c is found to be -10 when x equals 25, with a total of six integer solutions for c and x. Among these, three solutions have both x and c as positive integers. The main inquiry shifts to determining the value of x when c is at its maximum. The conversation indicates a collaborative effort to explore these mathematical relationships for a puzzle site.
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a = x^2 + 2cx + 1
b = 2x + 3c + 3

a / b = 5 plus a remainder of 11.

c=?

Want to include above in a "puzzle" site.
Want see if it makes sense..

Can you solve it?
Than you.
 
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Wilmer said:
...Can you solve it?...

Yes:

One easy solution is:

$$c=-10$$

I will be glad to show what I did if required.
 
MarkFL said:
Yes:
One easy solution is:
$$c=-10$$
I will be glad to show what I did if required.
Yippee! Yes, c=-10 (x=25) is a solution.

There are 6 solutions (I'm 99% sure) where c and x are integers (positive or negative).
3 of those have x and c > 0.

Hokay: question is now: what is x when c is at maximum?

Thanks Mark. Nice to hear from you !
 

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