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The problem is asking to select the equation that is equivalent to the given equation:

\[
x^2 + 16x = 22
\]

We need to complete the square to rewrite this equation in a form that matches the options provided.

### Step 1: Move the constant to the other side.
\[
x^2 + 16x = 22
\]
Now, let's focus on completing the square.

### Step 2: Complete the square.
To complete the square, take half of the coefficient of \(x\), which is 16, divide it by 2 to get 8, and then square it:
\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\]
Add this value to both sides of the equation:
\[
x^2 + 16x + 64 = 22 + 64
\]
\[
(x + 8)^2 = 86
\]

### Step 3: Compare with the given options.
The equation becomes \((x + 8)^2 = 86\), which matches option **C**.

### Final Answer:
The correct answer is **C**.

Would you like more details or have any questions? Here are five additional related questions:

1. What does "completing the square" mean in solving quadratic equations?
2. How can we solve the equation \((x + 8)^2 = 86\)?
3. Why do we divide the linear term's coefficient by 2 when completing the square?
4. What are the differences between factored form and completed square form?
5. How do we convert a quadratic equation from standard form to vertex form?

**Tip**: Completing the square is a powerful technique for solving quadratic equations, especially when factoring is difficult. It also helps when working with quadratic functions in graphing.

Which equation is equivalent to the given equation: x^2 + 16x = 22?

Learn how to solve the quadratic equation x^2 + 16x = 22 using the method of completing the square. Step-by-step explanation suitable for Grades 8-10, including the transformation to vertex form and comparing equivalent equations.

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