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To solve the given quadratic equation by completing the square, we can follow these steps:

The given equation is:
\[ -34 = x^2 - 14x + 10 \]

1. **Rewrite the equation in standard form**:
\[ x^2 - 14x + 10 + 34 = 0 \]
\[ x^2 - 14x + 44 = 0 \]

2. **Move the constant term to the right side**:
\[ x^2 - 14x = -44 \]

3. **Complete the square**:
To complete the square, we need to add and subtract a term that makes the left-hand side a perfect square trinomial. The term to add and subtract is \(\left(\frac{-14}{2}\right)^2 = 49\).

\[ x^2 - 14x + 49 - 49 = -44 \]
\[ (x - 7)^2 - 49 = -44 \]
\[ (x - 7)^2 = 5 \]

4. **Solve for \( x \)**:
\[ x - 7 = \pm \sqrt{5} \]
\[ x = 7 \pm \sqrt{5} \]

So, the solutions to the equation are:
\[ x = 7 + \sqrt{5} \]
\[ x = 7 - \sqrt{5} \]

Thus, the correct answer is **C: \( x = 7 + \sqrt{5} \)**.

Would you like more details or have any questions?

Here are 8 related questions to deepen your understanding:
1. How do you determine the term needed to complete the square?
2. What is the importance of completing the square in solving quadratic equations?
3. How can you verify the solutions of a quadratic equation?
4. What are some other methods to solve quadratic equations?
5. Can all quadratic equations be solved by completing the square?
6. What is the quadratic formula and how does it relate to completing the square?
7. How does completing the square help in graphing quadratic functions?
8. What are some real-world applications of solving quadratic equations?

**Tip:** When solving quadratic equations, always double-check your steps to ensure accuracy, especially when completing the square or using the quadratic formula.

Learn how to solve the quadratic equation x^2 - 14x + 44 by completing the square with a detailed step-by-step solution. This problem covers key algebraic concepts, including completing the square and the quadratic formula. Suitable for students in Grades 10-12.

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