To address this problem:

Identifying the Error

The student made a mistake in completing the square, as described in option C:

• "The student did not add the value for completing the square to both sides of the equation."

Explanation

Here's how to correctly solve this equation by completing the square:

1. Starting Equation: $0 = x^2 + 16x - 5$
2. Move the Constant to the Other Side: $x^2 + 16x = 5$
3. Complete the Square:
   • Take half of the coefficient of $x$, which is $16$, divide by 2 to get $8$, and then square it to get $64$.
   • Add $64$ to both sides of the equation to maintain equality: $x^2 + 16x + 64 = 5 + 64$ $x^2 + 16x + 64 = 69$
4. Rewrite as a Perfect Square: $(x + 8)^2 = 69$
5. Solve for $x$:
   • Take the square root of both sides, remembering to consider both the positive and negative roots: $x + 8 = \pm \sqrt{69}$
   • Isolate $x$: $x = -8 \pm \sqrt{69}$

Final Answer

The solutions for $x$ are: $x = -8 + \sqrt{69} \quad \text{and} \quad x = -8 - \sqrt{69}$

Let me know if you have any questions or need further clarification!

Additional Questions

1. What does completing the square mean in the context of solving quadratics?
2. Why is it necessary to add the same value to both sides when completing the square?
3. How would the solution change if the equation had a different constant term?
4. Can you use the quadratic formula to solve this equation instead, and how would it compare?
5. Why is it important to consider both positive and negative roots when solving quadratic equations?

Tip

When completing the square, always double-check that you add the square of half the $x$-coefficient to both sides to keep the equation balanced.