We are given the equation \( x = 3 - 2\sqrt{2} \). The goal is to find:

\[
\frac{\sqrt{x + 1}}{\sqrt{x}}
\]

Let's proceed step by step:

### Step 1: Simplify \( x + 1 \)

We know that \( x = 3 - 2\sqrt{2} \). So,

\[
x + 1 = (3 - 2\sqrt{2}) + 1 = 4 - 2\sqrt{2}
\]

### Step 2: Find the square roots of \( x \) and \( x + 1 \)

We need to find the square roots of both \( x = 3 - 2\sqrt{2} \) and \( x + 1 = 4 - 2\sqrt{2} \).

#### Square root of \( x \):
\[
x = 3 - 2\sqrt{2} = (\sqrt{2} - 1)^2
\]
So,
\[
\sqrt{x} = \sqrt{3 - 2\sqrt{2}} = \sqrt{(\sqrt{2} - 1)^2} = \sqrt{2} - 1
\]

#### Square root of \( x + 1 \):
\[
x + 1 = 4 - 2\sqrt{2} = (\sqrt{2} + 1)^2
\]
So,
\[
\sqrt{x + 1} = \sqrt{4 - 2\sqrt{2}} = \sqrt{(\sqrt{2} + 1)^2} = \sqrt{2} + 1
\]

### Step 3: Calculate \( \frac{\sqrt{x + 1}}{\sqrt{x}} \)

Now we calculate the expression:

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

To simplify this, multiply the numerator and denominator by \( \sqrt{2} + 1 \) (the conjugate of the denominator):

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

Simplifying the numerator and denominator:

- Numerator: \( (\sqrt{2} + 1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \)
- Denominator: \( (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \)

Thus, we get:

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

So, the final answer is:

\[
\boxed{3 + 2\sqrt{2}}
\]

Would you like more details on any step or have questions? Here are 5 related questions:

1. How do you simplify expressions with square roots in the numerator and denominator?
2. What is the process of rationalizing denominators?
3. How can we simplify expressions involving binomials like \( (\sqrt{2} - 1)^2 \)?
4. Why do we use conjugates when simplifying square root expressions?
5. Can you explain how \( \sqrt{3 - 2\sqrt{2}} \) simplifies to \( \sqrt{2} - 1 \)?

**Tip**: Always check for patterns such as perfect squares when simplifying complex square root expressions.

Learn how to simplify √(x + 1) / √x for x = 3 - 2√2 step-by-step. This problem involves simplifying radicals and rationalizing denominators, using advanced algebraic techniques. Suitable for students in Grades 10-12.

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