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The question asks to "Determine a number that must be added to make each of the following a perfect square trinomial."

Let's break down the problems step by step.

### 1. \( x^2 - 6x \)
To complete the square:
- Take half of the coefficient of \(x\) (which is -6), divide by 2 to get -3.
- Then square this value: \( (-3)^2 = 9 \).
- The number that must be added is **9**.

### 2. \( x^2 + 10x \)
To complete the square:
- Take half of the coefficient of \(x\) (which is 10), divide by 2 to get 5.
- Then square this value: \( 5^2 = 25 \).
- The number that must be added is **25**.

### 3. \( x^2 - 3x \)
To complete the square:
- Take half of the coefficient of \(x\) (which is -3), divide by 2 to get \(-\frac{3}{2}\).
- Then square this value: \( \left(-\frac{3}{2}\right)^2 = \frac{9}{4} \).
- The number that must be added is **\(\frac{9}{4}\)**.

### 4. \( x^2 + 2x \)
To complete the square:
- Take half of the coefficient of \(x\) (which is 2), divide by 2 to get 1.
- Then square this value: \( 1^2 = 1 \).
- The number that must be added is **1**.

Would you like more details on any of these steps or do you have any questions?

### Related Questions:
1. How do you complete the square for a quadratic expression with a leading coefficient other than 1?
2. What is the relationship between completing the square and the quadratic formula?
3. How can completing the square be used to derive the vertex form of a quadratic equation?
4. What is the geometric interpretation of completing the square?
5. How does completing the square help solve quadratic equations?

### Tip:
When completing the square, always ensure that the quadratic term has a coefficient of 1. If not, factor out the leading coefficient first before completing the square.

Determine a number that must be added to make each of the following a perfect square trinomial.

Learn how to determine the number needed to complete the square for quadratic expressions. This step-by-step guide covers essential algebraic concepts and is suitable for students in Grades 7-9.

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