https://file.aiquickdraw.com/mathbot/file/17266269488319kvrm74n.jpg

The problem asks you to solve the quadratic equation \( x^2 + 8x + 11 = 0 \) by completing the square. Let me walk you through the steps.

### Step 1: Move the constant term to the other side
We start with the equation:
\[
x^2 + 8x + 11 = 0
\]
Subtract 11 from both sides:
\[
x^2 + 8x = -11
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( x \) (which is 8), divide it by 2, and square it:
\[
\left(\frac{8}{2}\right)^2 = 16
\]
Add 16 to both sides of the equation:
\[
x^2 + 8x + 16 = -11 + 16
\]
This simplifies to:
\[
(x + 4)^2 = 5
\]

### Step 3: Solve by taking the square root
Now, take the square root of both sides:
\[
x + 4 = \pm \sqrt{5}
\]

### Step 4: Solve for \( x \)
Subtract 4 from both sides to isolate \( x \):
\[
x = -4 \pm \sqrt{5}
\]

Thus, the solutions are:
\[
x = -4 + \sqrt{5}, \quad x = -4 - \sqrt{5}
\]

### Final Answer
- The equation in completed square form is: \( (x + 4)^2 = 5 \)
- The solutions are: \( x = -4 + \sqrt{5}, -4 - \sqrt{5} \)

Let me know if you'd like more details or have questions. 

Here are 5 related questions to deepen your understanding:
1. What is the general formula for completing the square?
2. How do you recognize when to use the method of completing the square?
3. Can you apply completing the square to an equation with a leading coefficient other than 1?
4. How is completing the square used to derive the quadratic formula?
5. What are the geometric interpretations of completing the square?

**Tip:** Completing the square is especially useful when solving quadratic equations that are not easily factorable.

Solve the quadratic equation by completing the square: x^2 + 8x + 11 = 0

Learn how to solve the quadratic equation x^2 + 8x + 11 = 0 using the completing the square method. This detailed solution breaks down the process step-by-step, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation