https://file.aiquickdraw.com/mathbot/file/1724866399347ucs2y8vp.jpg

Thank you for your patience. The correct interpretation of the equation from the image is:

\[
3x + 2\sqrt{9 + x^2} = 0
\]

Let's solve this equation step by step.

### Step 1: Isolate the square root term
To start, move the \(3x\) term to the other side of the equation:

\[
2\sqrt{9 + x^2} = -3x
\]

### Step 2: Square both sides
To eliminate the square root, square both sides of the equation:

\[
\left(2\sqrt{9 + x^2}\right)^2 = (-3x)^2
\]

This simplifies to:

\[
4(9 + x^2) = 9x^2
\]

### Step 3: Expand and rearrange
Expand the left side:

\[
36 + 4x^2 = 9x^2
\]

Move all terms involving \(x^2\) to one side:

\[
36 = 5x^2
\]

### Step 4: Solve for \(x^2\)
Divide both sides by 5 to isolate \(x^2\):

\[
x^2 = \frac{36}{5}
\]

\[
x^2 = 7.2
\]

### Step 5: Solve for \(x\)
Take the square root of both sides:

\[
x = \pm \sqrt{7.2}
\]

\[
x \approx \pm 2.68
\]

So, the solutions for \(x\) are approximately \( x \approx 2.68 \) and \( x \approx -2.68 \).

### Step 6: Verify the solutions
Substitute these values back into the original equation to ensure they are valid solutions.

Would you like me to verify these solutions or explain any step in more detail?

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**Related Questions:**
1. What are the potential pitfalls when solving equations involving square roots?
2. How do you simplify square roots like \(\sqrt{7.2}\)?
3. What are extraneous solutions, and how can they arise in this context?
4. How can we approximate square roots without a calculator?
5. What is the significance of the ± sign when solving quadratic equations?

**Tip:** Always check both positive and negative solutions when taking square roots, as both might be valid depending on the original equation.

Learn how to solve the equation 3x + 2√(9 + x^2) = 0 with a detailed step-by-step solution. This problem involves algebraic manipulation and solving equations with square roots. Suitable for students in Grades 10-12.

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