Given the equation \( a^2 + b^2 = 2809 \), there are multiple pairs of values for \( a \) and \( b \) that can satisfy this equation. To find integer values for \( a \) and \( b \), we can start by factoring 2809, or checking values for \( a \) and solving for \( b \).

1. Take a value of \( a \), for example \( a = 35 \).
   
   \[
   a^2 = 35^2 = 1225
   \]

2. Substitute this into the original equation:

   \[
   a^2 + b^2 = 2809 \implies 1225 + b^2 = 2809
   \]

   \[
   b^2 = 2809 - 1225 = 1584
   \]

Since \( b^2 = 1584 \), this is not a perfect square, so \( a = 35 \) is not a solution. We continue testing different integer values for \( a \).

Trying another value, say \( a = 37 \):

\[
a^2 = 37^2 = 1369
\]

Substitute into the equation:

\[
1369 + b^2 = 2809
\]

\[
b^2 = 2809 - 1369 = 1440
\]

Again, 1440 is not a perfect square.

Now, let's try \( a = 53 \):

\[
a^2 = 53^2 = 2809
\]

In this case:

\[
2809 + b^2 = 2809 \implies b^2 = 0
\]

Thus, \( b = 0 \), and the solution is \( a = 53 \), \( b = 0 \).

Therefore, one possible solution is \( a = 53 \) and \( b = 0 \).

Would you like to explore other solutions or have any questions?

Here are 5 related questions to expand on this concept:

1. What are the conditions for a number to be a perfect square?
2. Can there be negative values for \( a \) or \( b \) in this problem?
3. How can we determine all pairs of integers that satisfy \( a^2 + b^2 = 2809 \)?
4. What is the significance of the Pythagorean theorem in equations like this?
5. How can we solve quadratic Diophantine equations in general?

**Tip**: When solving equations like \( a^2 + b^2 = N \), it's helpful to check for perfect squares and use factoring techniques!

Discover how to find integer values for a and b in the equation a^2 + b^2 = 2809. This problem involves algebraic concepts such as perfect squares and quadratic Diophantine equations. Suitable for students in Grades 9-12.

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