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Let's analyze the question. You are being asked to match the incomplete quadratic equations with their corresponding perfect square forms. We have three perfect square forms listed as:

- \( (y + 5)^2 \)
- \( (y - 12)^2 \)
- \( (y + 15)^2 \)
- \( (y - 15)^2 \)

### Step-by-step approach:

#### 1. First equation: \( y^2 + 10y + ? \)
To complete the square, use the middle term \(10y\):
- Divide the coefficient of \(y\) by 2: \( \frac{10}{2} = 5 \).
- Square this result: \( 5^2 = 25 \).
Thus, \( y^2 + 10y + 25 = (y + 5)^2 \).

#### 2. Second equation: \( y^2 - 24y + ? \)
- Divide the coefficient of \(y\) by 2: \( \frac{-24}{2} = -12 \).
- Square this result: \( (-12)^2 = 144 \).
Thus, \( y^2 - 24y + 144 = (y - 12)^2 \).

#### 3. Third equation: \( y^2 + 30y + ? \)
- Divide the coefficient of \(y\) by 2: \( \frac{30}{2} = 15 \).
- Square this result: \( 15^2 = 225 \).
Thus, \( y^2 + 30y + 225 = (y + 15)^2 \).

### Matching the results:
- \( y^2 + 10y + ? \) matches \( (y + 5)^2 \)
- \( y^2 - 24y + ? \) matches \( (y - 12)^2 \)
- \( y^2 + 30y + ? \) matches \( (y + 15)^2 \)

Would you like to go into more detail or have any questions about these steps?

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Here are 5 additional questions related to this topic:
1. How do you complete the square for a quadratic equation with negative coefficients?
2. What is the geometric interpretation of a perfect square trinomial?
3. How can you factor quadratic equations that are not perfect squares?
4. What is the purpose of completing the square when solving quadratic equations?
5. How does completing the square help in deriving the quadratic formula?

**Tip:** When completing the square, always ensure you divide the coefficient of the linear term by 2 before squaring it to find the constant.

Consider the incomplete quadratic equation given in the first column. Select their corresponding perfect square equation.

Learn how to match incomplete quadratic equations with their corresponding perfect square forms. This problem involves completing the square, a key algebraic technique in quadratic equations. Suitable for students in Grades 8-10.

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