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We are given the equation:

\[
\frac{\sqrt{2} - 1}{\sqrt{2} + 1} = a + b\sqrt{2}
\]

To solve for \(a\) and \(b\), let's simplify the left-hand side first by rationalizing the denominator.

### Simplifying the Expression

Multiply the numerator and the denominator by the conjugate of the denominator:

\[
\frac{(\sqrt{2} - 1)(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{(\sqrt{2} - 1)^2}{2 - 1}
\]

Expanding and simplifying the numerator:

\[
(\sqrt{2} - 1)^2 = \sqrt{2}^2 - 2 \cdot \sqrt{2} \cdot 1 + 1^2 = 2 - 2\sqrt{2} + 1
\]
\[
= 3 - 2\sqrt{2}
\]

Thus, we have:

\[
\frac{3 - 2\sqrt{2}}{1} = 3 - 2\sqrt{2}
\]

### Matching the Form \(a + b\sqrt{2}\)

Now, equate this to \(a + b\sqrt{2}\):

\[
3 - 2\sqrt{2} = a + b\sqrt{2}
\]

From this, we match the coefficients:

- Coefficient of \(\sqrt{2}\): \(b = -2\)
- Constant term: \(a = 3\)

### Conclusion

Thus, the values of \(a\) and \(b\) are:
\[
a = 3, \quad b = -2
\]

Would you like to explore further details or have any questions about the steps taken?

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Further Exploration:
1. What happens if you multiply by a different conjugate?
2. How does this form relate to the standard form of complex numbers?
3. What other algebraic identities could help in similar problems?
4. Can this technique be used to simplify other irrational expressions?
5. What is the geometric interpretation of rationalizing the denominator?

**Tip**: When matching terms in algebraic equations, align like terms vertically for clarity and ease of comparison.

Find the value of a and b such that (√2 - 1)/(√2 + 1) = a + b√2

Learn how to find the values of a and b such that (√2 - 1)/(√2 + 1) equals a + b√2. This problem involves rationalizing denominators and simplifying square root expressions, suitable for students in Grades 9-11.

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