I. ` p^(2) - 8p+15 = 0`
II. ` q^(2) - 5q =- 6`

Options

A. if`p lt q,`
B. if`p gt q,`
C. if` p le q,`
D. if` p ge q,`

Explanation

To solve the given equations step by step, we will first solve for \( p \) in the first equation and then for \( q \) in the second equation.

### Step 1: Solve the first equation \( p^2 - 8p + 15 = 0 \)

1. **Identify the equation**:
\[ p^2 - 8p + 15 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 15 \) (the constant term) and add up to \( -8 \) (the coefficient of \( p \)).
The numbers are \( -3 \) and \( -5 \).

Thus, we can factor the equation as:
\[ (p - 3)(p - 5) = 0 \]

3. **Set each factor to zero**:
\[ p - 3 = 0 \quad \text{or} \quad p - 5 = 0 \]

4. **Solve for \( p \)**:
\[ p = 3 \quad \text{or} \quad p = 5 \]

### Step 2: Solve the second equation \( q^2 - 5q = -6 \)

1. **Rearrange the equation**:
\[ q^2 - 5q + 6 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 6 \) and add up to \( -5 \).
The numbers are \( -2 \) and \( -3 \).

Thus, we can factor the equation as:
\[ (q - 2)(q - 3) = 0 \]

3. **Set each factor to zero**:
\[ q - 2 = 0 \quad \text{or} \quad q - 3 = 0 \]

4. **Solve for \( q \)**:
\[ q = 2 \quad \text{or} \quad q = 3 \]

### Step 3: Determine the relationship between \( p \) and \( q \)

From the solutions we found:
- Possible values for \( p \) are \( 3 \) and \( 5 \).
- Possible values for \( q \) are \( 2 \) and \( 3 \).

Now we analyze the relationships:
1. If \( p = 3 \), then \( q = 3 \) (they are equal).
2. If \( p = 5 \), then \( q = 2 \) (here \( p > q \)).

Thus, we can conclude:
- The relationship can be expressed as \( p \geq q \).

### Final Answer:
The values of \( p \) are \( 3 \) and \( 5 \), and the values of \( q \) are \( 2 \) and \( 3 \). The relationship between \( p \) and \( q \) is \( p \geq q \).

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