https://file.aiquickdraw.com/mathbot/file/1725469574247goizgdm9.jpg

https://file.aiquickdraw.com/mathbot/file/1725469574249jrs5sios.jpg

To solve the problem and determine whether Option 'A' or Option 'B' (or neither) is correct, let's first complete the square for the given function:

### Given Equation:
\[ 4x^2 + 8x = 13 \]

### Step 1: Factor out the coefficient of \( x^2 \)
Since the equation is in the form of \( 4x^2 + 8x \), we can factor out the 4:
\[
4(x^2 + 2x) = 13
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( x \) (which is 2 in this case), square it, and add it inside the parentheses:
\[
x^2 + 2x + 1 = (x + 1)^2
\]

However, since we added 1 inside the parentheses, we actually added \( 4 \times 1 = 4 \) to the left side of the equation (because of the factor 4 outside the parentheses). To balance the equation, we must add 4 to the right side as well.

Thus, the equation becomes:
\[
4(x + 1)^2 = 13 + 4 = 17
\]

### Step 3: Simplify the equation
Now the equation is:
\[
4(x + 1)^2 = 17
\]
We can compare this with the options provided.

### Option 'A':
- **Row 1**: The value that must be added to complete the square is \( 4 \), which is correct.
- **Row 2**: The equivalent equation is \( 4(x + 1)^2 = 17 \), which is also correct.
- **Row 3**: The solution can be found by solving \( 4(x + 1)^2 = 17 \):
  \[
  (x + 1)^2 = \frac{17}{4}
  \]
  Taking the square root of both sides:
  \[
  x + 1 = \pm \sqrt{\frac{17}{4}} = \pm \frac{\sqrt{17}}{2}
  \]
  Therefore:
  \[
  x = -1 \pm \frac{\sqrt{17}}{2}
  \]
  This corresponds to Option 'A', which shows:
  \[
  x = -1 \pm \frac{\sqrt{17}}{2} \quad \text{or} \quad x = -1 \pm \frac{3}{2}
  \]

### Option 'B':
- **Row 1**: The value that must be added to complete the square is listed as \( 16 \), which is incorrect.
- **Row 2**: The equivalent equation is \( 4(x + 1)^2 = 9 \), which is incorrect.
- **Row 3**: The solution expression is incorrect.

### Conclusion:
**Option 'A'** is the correct choice.

---

Do you have any questions or would you like more details? 

Here are some related questions you might find useful:
1. What are the general steps for completing the square?
2. How does completing the square help in solving quadratic equations?
3. Why is it necessary to balance both sides of an equation when adding or subtracting terms?
4. What are the benefits of solving a quadratic equation by completing the square instead of factoring?
5. Can completing the square be used for any quadratic equation, and are there cases where it’s more efficient than the quadratic formula?

**Tip:** Always double-check each step when completing the square, especially when balancing both sides of the equation. This helps avoid errors that could lead to incorrect solutions.

Answer the following questions regarding completing the square for the function: 4x^2 + 8x = 13. Select Option 'A' or Option 'B', or neither.

Learn how to solve the quadratic equation 4x^2 + 8x = 13 by completing the square. This step-by-step guide walks through factoring, balancing the equation, and determining the correct solution. Perfect for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation