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To solve the equation \(3x^2 + 21 = 0\), follow these steps:

1. **Rewrite the equation:**  
   \[ 3x^2 + 21 = 0 \]

2. **Move the constant term to the right side:**  
   \[ 3x^2 = -21 \]

3. **Divide both sides by 3:**  
   \[ x^2 = -7 \]

4. **Take the square root of both sides:**  
   \[ x = \pm \sqrt{-7} \]

5. **Simplify using imaginary numbers:**  
   Since \(\sqrt{-7} = \sqrt{7}i\), the solutions are:  
   \[ x = \pm \sqrt{7}i \]

Thus, the solutions for \(x\) are \( x = \sqrt{7}i \) and \( x = -\sqrt{7}i \).

Would you like further details on this solution, or do you have any other questions?

Here are five related questions:
1. How do you solve quadratic equations with real solutions?
2. What are imaginary numbers, and how do they relate to quadratic equations?
3. How can you check if a quadratic equation has real or complex solutions?
4. What are some practical applications of imaginary numbers?
5. How does the quadratic formula work for solving quadratic equations?

**Tip:** When solving equations that result in a negative number under a square root, the solutions will be complex, involving imaginary numbers.

Solve the quadratic equation 3x^2 + 21 = 0 and learn how to work with complex roots using the imaginary unit i. Follow a step-by-step guide that simplifies the solution. Suitable for students in Grades 9-11.

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