To multiply the expression \(\sqrt{x} - \sqrt{7}\) by its conjugate, we first identify the conjugate of the expression.

### Step-by-Step Solution

1. **Identify the conjugate:**
   - The conjugate of \(\sqrt{x} - \sqrt{7}\) is \(\sqrt{x} + \sqrt{7}\).

2. **Multiply the expression by its conjugate:**
   - \[
   (\sqrt{x} - \sqrt{7})(\sqrt{x} + \sqrt{7})
   \]

3. **Use the difference of squares formula:**
   - The product of a binomial and its conjugate is the difference of squares:
   - \[
   (a - b)(a + b) = a^2 - b^2
   \]

   Applying this formula:
   - \[
   (\sqrt{x})^2 - (\sqrt{7})^2
   \]

4. **Simplify each square:**
   - \[
   (\sqrt{x})^2 = x \quad \text{and} \quad (\sqrt{7})^2 = 7
   \]

5. **Combine and simplify:**
   - \[
   x - 7
   \]

Thus, the result of multiplying \(\sqrt{x} - \sqrt{7}\) by its conjugate is:

\[
(\sqrt{x} - \sqrt{7})(\sqrt{x} + \sqrt{7}) = x - 7.
\]

Would you like to explore more about multiplying conjugates, or do you have any other questions?

### Related Questions:

1. How do you simplify expressions involving square roots and conjugates?
2. What is the purpose of multiplying by the conjugate in a mathematical context?
3. How do you handle conjugates in complex numbers?
4. What happens if the terms inside the square roots are variables rather than constants?
5. Can you multiply conjugates in fractions to simplify expressions?

**Tip:** Always remember that the product of conjugates eliminates the square roots by resulting in a difference of squares, making the expression easier to handle.

Learn how to multiply the conjugate of √x - √7 using the difference of squares formula. This problem involves algebraic manipulation with square roots and conjugates. Suitable for high school students.

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