Answer key for completing the square problems, showing values of "c" for quadratic expressions.

Problem Overview:

The task involves completing the square for quadratic expressions of the form \( ax^2 + bx + c \) and determining the value of \( c \) that makes the expression a perfect square trinomial. The general method for completing the square is as follows:

1. Start with a quadratic expression in the form \( x^2 + bx + c \).
2. Identify the coefficient of \( x \), which is \( b \).
3. Compute \( \left( \frac{b}{2} \right)^2 \). This value is the constant term \( c \) that completes the square.
4. Add this value to the expression to make it a perfect square trinomial.

Solution for Each Problem:

#### 1) \( y^2 - 8y + c \)
- Coefficient of \( y \): \( -8 \)
- Compute \( \left( \frac{-8}{2} \right)^2 = (-4)^2 = 16 \)
- Therefore, \( c = 16 \)

#### 2) \( h^2 + 24h + c \)
- Coefficient of \( h \): \( 24 \)
- Compute \( \left( \frac{24}{2} \right)^2 = (12)^2 = 144 \)
- Therefore, \( c = 144 \)

#### 3) \( d^2 - 17d + c \)
- Coefficient of \( d \): \( -17 \)
- Compute \( \left( \frac{-17}{2} \right)^2 = \left( -\frac{17}{2} \right)^2 = \frac{289}{4} \)
- Therefore, \( c = \frac{289}{4} \)

#### 4) \( m^2 + 12m + c \)
- Coefficient of \( m \): \( 12 \)
- Compute \( \left( \frac{12}{2} \right)^2 = (6)^2 = 36 \)
- Therefore, \( c = 36 \)

#### 5) \( r^2 + 15r + c \)
- Coefficient of \( r \): \( 15 \)
- Compute \( \left( \frac{15}{2} \right)^2 = \left( \frac{15}{2} \right)^2 = \frac{225}{4} \)
- Therefore, \( c = \frac{225}{4} \)

#### 6) \( k^2 - 5k + c \)
- Coefficient of \( k \): \( -5 \)
- Compute \( \left( \frac{-5}{2} \right)^2 = \left( -\frac{5}{2} \right)^2 = \frac{25}{4} \)
- Therefore, \( c = \frac{25}{4} \)

#### 7) \( y^2 + 3y + c \)
- Coefficient of \( y \): \( 3 \)
- Compute \( \left( \frac{3}{2} \right)^2 = \left( \frac{3}{2} \right)^2 = \frac{9}{4} \)
- Therefore, \( c = \frac{9}{4} \)

#### 8) \( x^2 - 9x + c \)
- Coefficient of \( x \): \( -9 \)
- Compute \( \left( \frac{-9}{2} \right)^2 = \left( -\frac{9}{2} \right)^2 = \frac{81}{4} \)
- Therefore, \( c = \frac{81}{4} \)

#### 9) \( n^2 - 10n + c \)
- Coefficient of \( n \): \( -10 \)
- Compute \( \left( \frac{-10}{2} \right)^2 = (-5)^2 = 25 \)
- Therefore, \( c = 25 \)

#### 10) \( h^2 - 16h + c \)
- Coefficient of \( h \): \( -16 \)
- Compute \( \left( \frac{-16}{2} \right)^2 = (-8)^2 = 64 \)
- Therefore, \( c = 64 \)

Final Answer:

\[
\boxed{
\begin{array}{ll}
1) & 16 \\
2) & 144 \\
3) & \frac{289}{4} \\
4) & 36 \\
5) & \frac{225}{4} \\
6) & \frac{25}{4} \\
7) & \frac{9}{4} \\
8) & \frac{81}{4} \\
9) & 25 \\
10) & 64 \\
\end{array}
}
\]