https://file.aiquickdraw.com/mathbot/file/1738590033630kugove9s.jpg

The equation presented in the image is:

\[
x + \frac{1}{x} = 0
\]
and we are asked to find the value of:

\[
\sqrt{2} \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)
\]

### Step 1: Solve for \( x \)
Starting with the equation \( x + \frac{1}{x} = 0 \):

\[
x + \frac{1}{x} = 0 \implies x = -\frac{1}{x}
\]

Multiplying both sides by \( x \) (assuming \( x \neq 0 \)):

\[
x^2 = -1
\]

This gives the solution \( x = i \) or \( x = -i \), where \( i \) is the imaginary unit.

### Step 2: Find the expression \( \sqrt{2} \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \)
Now, we need to compute \( \sqrt{2} \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \) for \( x = i \) (since \( x = -i \) gives the same result).

Let’s find the values of \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \) for \( x = i \):

- \( \sqrt{i} \) is a complex number, and its value is \( \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \).
- \( \frac{1}{\sqrt{i}} \) will be the reciprocal of this, which is \( \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \).

Adding these two gives:

\[
\sqrt{i} + \frac{1}{\sqrt{i}} = \left( \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \right) + \left( \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \right)
\]
This simplifies to:

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

### Step 3: Final answer
Now multiplying by \( \sqrt{2} \):

\[
\sqrt{2} \left( \sqrt{i} + \frac{1}{\sqrt{i}} \right) = \sqrt{2} \times \sqrt{2} = 2
\]

Thus, the value of the given expression is \( \boxed{2} \).

Would you like further clarification or details on any steps?

---

Here are some additional questions related to this topic:

1. How would you solve for \( x \) if the equation was \( x^2 + \frac{1}{x^2} = 0 \)?
2. What is the general solution for equations involving complex numbers like \( x + \frac{1}{x} = 0 \)?
3. How do you simplify expressions like \( \sqrt{x} + \frac{1}{\sqrt{x}} \) for other values of \( x \)?
4. Can this method be used for solving real number equations, or is it strictly for complex numbers?
5. How do we handle complex numbers in square roots and fractional powers?

Tip: When dealing with complex numbers in expressions, it's often helpful to use polar form for easier computation and simplification.

Given that x + 1/x = 0, find the value of √2(√x + 1/√x).

Learn how to solve the equation x + 1/x = 0 and determine the value of √2(√x + 1/√x) using complex numbers and radical properties. This problem explores algebra and complex number theory, making it ideal for high school students.

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