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The given expression is:

\[
\frac{3 - \sqrt{n}}{2 + \sqrt{n}}
\]

It is stated that this expression can be written in the form:

\[
a + b\sqrt{n}
\]

where \( a \) and \( b \) are rational, and \( \sqrt{n} \) is irrational. We need to find expressions for \( a \) and \( b \) in terms of \( n \).

### Step-by-step solution:

1. **Multiply numerator and denominator by the conjugate**:
   To eliminate the square root in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, \( 2 - \sqrt{n} \).

   \[
   \frac{3 - \sqrt{n}}{2 + \sqrt{n}} \times \frac{2 - \sqrt{n}}{2 - \sqrt{n}} = \frac{(3 - \sqrt{n})(2 - \sqrt{n})}{(2 + \sqrt{n})(2 - \sqrt{n})}
   \]

2. **Simplify the denominator**:
   The denominator is a difference of squares, so:

   \[
   (2 + \sqrt{n})(2 - \sqrt{n}) = 2^2 - (\sqrt{n})^2 = 4 - n
   \]

3. **Simplify the numerator**:
   Expand the numerator using the distributive property:

   \[
   (3 - \sqrt{n})(2 - \sqrt{n}) = 3(2) + 3(-\sqrt{n}) - \sqrt{n}(2) - \sqrt{n}(-\sqrt{n})
   \]

   Simplifying each term:

   \[
   = 6 - 3\sqrt{n} - 2\sqrt{n} + n = 6 - 5\sqrt{n} + n
   \]

4. **Put it all together**:
   The expression becomes:

   \[
   \frac{6 - 5\sqrt{n} + n}{4 - n}
   \]

5. **Separate the expression**:
   Now, we separate this into two parts to identify the coefficients of the rational and irrational parts:

   \[
   \frac{6 + n}{4 - n} - \frac{5\sqrt{n}}{4 - n}
   \]

   Thus, we can identify:

   \[
   a = \frac{6 + n}{4 - n} \quad \text{and} \quad b = \frac{-5}{4 - n}
   \]

### Final Answer:

The expression can be written as:

\[
a + b\sqrt{n} = \frac{6 + n}{4 - n} - \frac{5}{4 - n}\sqrt{n}
\]

where \( a = \frac{6 + n}{4 - n} \) and \( b = \frac{-5}{4 - n} \).

---

Would you like further details or clarification on any of these steps? Here are some related questions you might find useful:

1. How do you handle expressions involving square roots in the denominator?
2. What is the significance of multiplying by the conjugate?
3. Can the difference of squares method be applied to other forms of irrational expressions?
4. How would you approach simplifying a more complex fraction with multiple square roots?
5. What happens if \( n \) is a specific value, like 4 or 9, in this problem?

**Tip**: Always check the denominator for any potential simplifications using known algebraic identities like the difference of squares!

The expression (3 - sqrt(n)) / (2 + sqrt(n)) can be written in the form a + b*sqrt(n), where a and b are rational but sqrt(n) is irrational. Find expressions for a and b in terms of n.

Learn how to simplify the expression (3 - sqrt(n)) / (2 + sqrt(n)) into the form a + b*sqrt(n), where a and b are rational and sqrt(n) is irrational. This problem involves the use of conjugates and the difference of squares method. Suitable for students in Grades 10-12.

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