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Let's work through the two questions one by one.

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**Question 1:**
Which value must be added to the expression \( x^2 - 8x \) to make it a perfect-square trinomial?

To make a quadratic expression \( x^2 + bx \) into a perfect-square trinomial, the value to be added is \( \left(\frac{b}{2}\right)^2 \).

For the expression \( x^2 - 8x \), the value of \( b \) is \(-8\).

1. First, divide \( b \) by 2:  
   \[
   \frac{-8}{2} = -4
   \]
2. Square the result:  
   \[
   (-4)^2 = 16
   \]

Thus, **16** must be added to the expression to make it a perfect-square trinomial.  
The correct answer is **16**.

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**Question 2:**
Which of the following expressions is a perfect square trinomial?

A perfect square trinomial has the form \( (x + a)^2 = x^2 + 2ax + a^2 \).

Let's check each option:

1. \( x^2 + 4x + 8 \):  
   The middle term is 4, so \( a \) should be \( \frac{4}{2} = 2 \).  
   But \( a^2 = 2^2 = 4 \), not 8. Therefore, this is not a perfect square trinomial.

2. \( x^2 + 8x + 15 \):  
   The middle term is 8, so \( a \) should be \( \frac{8}{2} = 4 \).  
   But \( a^2 = 4^2 = 16 \), not 15. This is not a perfect square trinomial.

3. \( x^2 - 10x + 25 \):  
   The middle term is -10, so \( a \) should be \( \frac{-10}{2} = -5 \).  
   \( a^2 = (-5)^2 = 25 \), which matches the constant term.  
   Thus, **this is a perfect square trinomial**.

4. \( x^2 + 6x + 12 \):  
   The middle term is 6, so \( a \) should be \( \frac{6}{2} = 3 \).  
   But \( a^2 = 3^2 = 9 \), not 12. This is not a perfect square trinomial.

The correct answer is \( x^2 - 10x + 25 \).

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Would you like further explanation on completing the square or identifying perfect-square trinomials?

Here are some additional questions for further practice:
1. How do you complete the square for a quadratic equation?
2. Can a quadratic expression with a negative constant term be a perfect square?
3. What is the general formula for factoring perfect-square trinomials?
4. How would you solve a quadratic equation using the method of completing the square?
5. What is the significance of the discriminant in determining whether a quadratic is factorable?

**Tip**: A quick way to recognize a perfect-square trinomial is to check if the constant term is the square of half the coefficient of the linear term.

Which value must be added to the expression x^2 - 8x to make it a perfect-square trinomial? Which of the following expressions is a perfect square trinomial?

Learn how to solve quadratic expressions, including completing the square for x^2 - 8x, and identifying perfect square trinomials such as x^2 - 10x + 25. This step-by-step guide is ideal for students in Grades 8-10 studying algebra and quadratic equations.

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