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The problem asks to complete the square for the quadratic expression \( x^2 - 6x + 78 \) in the form \( (x + A)^2 + B \).

Let's go step by step to complete the square:

1. **Starting expression**:  
   \( x^2 - 6x + 78 \)

2. **Factor out the coefficient of \( x^2 \) (which is 1 here, so nothing changes)**:  
   \( x^2 - 6x + 78 \)

3. **Take half of the coefficient of \( x \) (which is -6)**:  
   \( \frac{-6}{2} = -3 \)

4. **Square this value**:  
   \( (-3)^2 = 9 \)

5. **Rewrite the expression by adding and subtracting this squared value**:  
   \( x^2 - 6x + 9 + 78 - 9 \)  
   This step doesn't change the expression because we are adding and subtracting 9 at the same time.

6. **Now factor the perfect square trinomial**:  
   \( (x - 3)^2 + 69 \)

Here, we have the expression in the form \( (x + A)^2 + B \). Therefore:
- \( A = -3 \)
- \( B = 69 \)

### Final answer:
- \( A = -3 \)
- \( B = 69 \)

Would you like a more detailed explanation on any part of this process?

### Follow-up Questions:
1. What is the general process for completing the square?
2. How do you factor a perfect square trinomial?
3. Why do we add and subtract the same number when completing the square?
4. How would this method change if there was a different coefficient for \( x^2 \)?
5. How can completing the square help in solving quadratic equations?

### Tip:
Completing the square is particularly useful for converting a quadratic equation into vertex form, which makes it easier to find the vertex of a parabola or solve the equation.

By completing the square, the expression x^2 - 6x + 78 equals (x + A)^2 + B where A = ? and B = ?

Solve the quadratic expression x^2 - 6x + 78 by completing the square to get the form (x + A)^2 + B. Learn how to find A and B through step-by-step guidance. Suitable for students in Grades 8-10.

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