I.` p^(2) + p = 56`
II.` q^(2) - 17q+ 72 = 0`

Options

A. if p is greater than q.
B. if p is smaller than q.
C. if p is equal q.
D. if p is either equal to or greater than q.

Explanation

To solve the equations given in the question, we will follow these steps:

### Step 1: Solve for \( p \) in the equation \( p^2 + p = 56 \)

First, we need to rearrange the equation into standard quadratic form:

\[ p^2 + p - 56 = 0 \]

### Step 2: Factor the quadratic equation

Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \(-56\) (the constant term) and add up to \(1\) (the coefficient of \(p\)).

The factors of \(-56\) that work are \(8\) and \(-7\) because:

\[ 8 \times (-7) = -56 \quad \text{and} \quad 8 + (-7) = 1 \]

Thus, we can factor the equation as:

\[ (p + 8)(p - 7) = 0 \]

### Step 3: Find the values of \( p \)

Now, we set each factor to zero:

1. \( p + 8 = 0 \) → \( p = -8 \)
2. \( p - 7 = 0 \) → \( p = 7 \)

So, the possible values for \( p \) are:

\[ p = -8 \quad \text{or} \quad p = 7 \]

### Step 4: Solve for \( q \) in the equation \( q^2 - 17q + 72 = 0 \)

Now, we will solve the second equation. This is already in standard form, so we can factor it directly.

### Step 5: Factor the quadratic equation for \( q \)

We need to find two numbers that multiply to \(72\) and add up to \(-17\). The factors of \(72\) that work are \(-8\) and \(-9\) because:

\[ (-8) \times (-9) = 72 \quad \text{and} \quad (-8) + (-9) = -17 \]

Thus, we can factor the equation as:

\[ (q - 8)(q - 9) = 0 \]

### Step 6: Find the values of \( q \)

Now, we set each factor to zero:

1. \( q - 8 = 0 \) → \( q = 8 \)
2. \( q - 9 = 0 \) → \( q = 9 \)

So, the possible values for \( q \) are:

\[ q = 8 \quad \text{or} \quad q = 9 \]

### Step 7: Compare the values of \( p \) and \( q \)

Now we have the values:

- For \( p \): \(-8\) and \(7\)
- For \( q \): \(8\) and \(9\)

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

Comparing the values:

- \( q = 8 \) is greater than \( p = 7 \)
- \( q = 9 \) is also greater than \( p = 7 \)
- \( q = 8 \) is greater than \( p = -8 \)
- \( q = 9 \) is also greater than \( p = -8 \)

Thus, we conclude that \( q \) is greater than \( p \).

### Final Answer

The relationship is:

\[ q > p \]