To solve the equation \( x^2 + \sqrt{2x} = 0 \), we proceed as follows:

### Step 1: Set up the equation
The equation is:
\[
x^2 + \sqrt{2x} = 0
\]

### Step 2: Isolate the square root term
Move the square root term to the other side of the equation:
\[
x^2 = -\sqrt{2x}
\]

### Step 3: Square both sides
To eliminate the square root, square both sides of the equation. However, since squaring both sides may introduce extraneous solutions, we'll need to check for them later.

First, square the left side:
\[
(x^2)^2 = x^4
\]

Now, square the right side:
\[
(-\sqrt{2x})^2 = 2x
\]

So the equation becomes:
\[
x^4 = 2x
\]

### Step 4: Solve the polynomial equation
Rearrange the equation to set it equal to zero:
\[
x^4 - 2x = 0
\]

Factor the equation:
\[
x(x^3 - 2) = 0
\]

### Step 5: Solve for \(x\)
Set each factor equal to zero:

1. \( x = 0 \)
2. \( x^3 - 2 = 0 \)

For the second equation:
\[
x^3 = 2
\]
\[
x = \sqrt[3]{2}
\]

### Step 6: Check for extraneous solutions
We need to check whether each solution satisfies the original equation:

1. **For \( x = 0 \)**:
   \[
   0^2 + \sqrt{2 \times 0} = 0 + 0 = 0
   \]
   This satisfies the original equation.

2. **For \( x = \sqrt[3]{2} \)**:
   \[
   (\sqrt[3]{2})^2 + \sqrt{2\sqrt[3]{2}}
   \]
   Calculating this shows it does not satisfy the original equation, so this is an extraneous solution.

### Final Answer:
The only solution is:
\[
x = 0
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you solve higher-degree polynomials?
2. What are extraneous solutions and how do they occur?
3. Can all polynomial equations be factored easily?
4. How do you check solutions in radical equations?
5. What is the difference between real and complex roots?
6. How do you solve cubic equations?
7. What are the properties of cubic and quartic equations?
8. How does the discriminant help in solving polynomial equations?

**Tip:** Always remember to check for extraneous solutions when solving equations involving radicals.

Learn how to solve the equation x^2 + sqrt(2x) = 0 with a detailed step-by-step solution. This problem covers equations involving radicals and polynomials, including the quadratic equation formula. Suitable for students in Grades 10-12.

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