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 !
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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