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We are given the following system of equations for \(a\) and \(b\):

\[
a^2 + 121 = b^2 + 1369 = a^2 + b^2 - 2ab + 576
\]

We need to solve for \(a\), \(b\), and ultimately find the value of \(ab\).

### Step 1: Analyze the First Equality
From the first equation:
\[
a^2 + 121 = b^2 + 1369
\]
Rearranging it:
\[
a^2 - b^2 = 1369 - 121 = 1248
\]
This can be factored as:
\[
(a - b)(a + b) = 1248
\]

### Step 2: Analyze the Second Equality
From the second equation:
\[
b^2 + 1369 = a^2 + b^2 - 2ab + 576
\]
Simplify it:
\[
1369 = a^2 - 2ab + 576
\]
Rearrange it:
\[
a^2 - 2ab = 1369 - 576 = 793
\]

### Step 3: Solve the System
Now we have the system:
\[
(a - b)(a + b) = 1248 \quad \text{(Equation 1)}
\]
\[
a^2 - 2ab = 793 \quad \text{(Equation 2)}
\]

We will solve this system step by step.

#### Want more details or clarification so far?

Here are 5 questions that expand the topic:
1. How can we solve for \(a\) and \(b\) using substitution?
2. What are the implications of factoring the difference of squares?
3. Could we find specific values of \(a\) and \(b\) using trial and error for integer solutions?
4. How does factoring simplify solving equations with higher powers?
5. What strategies can we use to simplify complex systems of equations?

**Tip:** When dealing with systems of equations, try to reduce the number of variables by substitution or elimination before solving.

Solve the following System of Equations for a and b: a^2 + 121 = b^2 + 1369 = a^2 + b^2 - 2ab + 576. Find the value of ab.

Learn how to solve a system of quadratic equations involving variables a and b. This step-by-step solution covers key algebraic techniques such as factoring and simplifying expressions. Suitable for high school students in Grades 10-12.

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