https://file.aiquickdraw.com/mathbot/file/17285933215386l5cfpc0.jpg

Let's solve the two questions from the image:

### **Question 3**: 
Solve by completing the square for \( x^2 + 10x = -81 \).

#### Steps:
1. **Move the constant term** to the other side:
   \[
   x^2 + 10x + \left( \frac{10}{2} \right)^2 = -81 + \left( \frac{10}{2} \right)^2
   \]
   Since \( \frac{10}{2} = 5 \), we add \( 5^2 = 25 \) to both sides:
   \[
   x^2 + 10x + 25 = -81 + 25
   \]
   \[
   x^2 + 10x + 25 = -56
   \]

2. **Write the left-hand side** as a square:
   \[
   (x + 5)^2 = -56
   \]

3. **Solve for \( x \)** by taking the square root of both sides:
   \[
   x + 5 = \pm \sqrt{-56}
   \]
   \[
   x + 5 = \pm \sqrt{56}i
   \]
   \[
   x + 5 = \pm 2i\sqrt{14}
   \]

4. **Solve for \( x \)**:
   \[
   x = -5 \pm 2i\sqrt{14}
   \]
   So, the solutions are:
   \[
   x = -5 + 2i\sqrt{14}, \quad x = -5 - 2i\sqrt{14}
   \]
   The correct answer is \( \boxed{-5 + 2i\sqrt{14}, -5 - 2i\sqrt{14}} \).

### **Question 4**:
Solve by completing the square for \( x^2 - 6x - 11 = 0 \).

#### Steps:
1. **Move the constant term** to the other side:
   \[
   x^2 - 6x = 11
   \]

2. **Complete the square**:
   Add \( \left( \frac{6}{2} \right)^2 = 9 \) to both sides:
   \[
   x^2 - 6x + 9 = 11 + 9
   \]
   \[
   (x - 3)^2 = 20
   \]

3. **Solve for \( x \)** by taking the square root of both sides:
   \[
   x - 3 = \pm \sqrt{20}
   \]
   \[
   x - 3 = \pm 2\sqrt{5}
   \]

4. **Solve for \( x \)**:
   \[
   x = 3 \pm 2\sqrt{5}
   \]
   The correct answer is \( \boxed{x = 3 \pm 2\sqrt{5}} \).

---

Let me know if you'd like more details or have any questions! Here are five relative questions you could explore:

1. Can you solve any quadratic equation using the completing the square method?
2. What are the advantages of using completing the square versus the quadratic formula?
3. How can you recognize when a quadratic equation will result in imaginary solutions?
4. How does completing the square relate to the vertex form of a quadratic equation?
5. Can completing the square be used to solve higher-degree polynomials?

**Tip**: Always check if the discriminant \( b^2 - 4ac \) is negative when solving quadratic equations to quickly determine if the solutions will involve imaginary numbers!

Solve by completing the square for x^2 + 10x = -81 and x^2 - 6x - 11 = 0.

Learn how to solve quadratic equations by completing the square. This step-by-step guide covers solving x^2 + 10x = -81 and x^2 - 6x - 11 = 0, exploring the algebraic process and imaginary solutions. Ideal for students in Grades 9-11.

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