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The question appears to ask for the minimum value of the expression:

\[
p^2 - 16pq + 64q^2 - 12
\]

We can simplify this expression by recognizing the terms \( p^2 - 16pq + 64q^2 \). This part resembles a perfect square. Let's rewrite it as follows:

\[
p^2 - 16pq + 64q^2 = (p - 8q)^2
\]

Therefore, the expression becomes:

\[
(p - 8q)^2 - 12
\]

Since \( (p - 8q)^2 \geq 0 \) for all real values of \( p \) and \( q \), the minimum value of \( (p - 8q)^2 \) is 0. Substituting 0 for \( (p - 8q)^2 \) gives:

\[
(p - 8q)^2 - 12 = 0 - 12 = -12
\]

Thus, the minimum value of the expression is \(-12\).

### Answer:
The correct option is **B) -12**.

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions for further practice:

1. How do you identify a perfect square in a quadratic expression?
2. What are the general steps to complete the square for a quadratic expression?
3. How can we find the minimum or maximum value of a quadratic expression?
4. What other methods can be used to simplify quadratic expressions?
5. How does the discriminant relate to finding the minimum or maximum value?

**Tip:** Recognizing perfect squares is a helpful skill for quickly simplifying and solving quadratic expressions.

Ifodaning eng kichik qiymatini toping: p^2 - 16pq + 64q^2 - 12

Learn how to find the minimum value of the expression p^2 - 16pq + 64q^2 - 12 using completing the square. This problem involves simplifying a quadratic expression and is suitable for students in Grades 9-10.

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