I. `p^(2) - 7p=- 12`
II. ` q^(2) - 3q+2=0`

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 and find the relationship between \( p \) and \( q \), we will follow these steps:

### Step 1: Solve the first equation for \( p \)
The first equation is:
\[ p^2 - 7p = -12 \]

To convert it into standard quadratic form, we add 12 to both sides:
\[ p^2 - 7p + 12 = 0 \]

### Step 2: Factor the quadratic equation
Now we will factor the equation:
\[ p^2 - 7p + 12 = (p - 3)(p - 4) = 0 \]

### Step 3: Find the values of \( p \)
Setting each factor to zero gives us:
1. \( p - 3 = 0 \) → \( p = 3 \)
2. \( p - 4 = 0 \) → \( p = 4 \)

Thus, the values of \( p \) are \( p = 3 \) and \( p = 4 \).

### Step 4: Solve the second equation for \( q \)
The second equation is:
\[ q^2 - 3q + 2 = 0 \]

### Step 5: Factor the quadratic equation
We will factor this equation:
\[ q^2 - 3q + 2 = (q - 1)(q - 2) = 0 \]

### Step 6: Find the values of \( q \)
Setting each factor to zero gives us:
1. \( q - 1 = 0 \) → \( q = 1 \)
2. \( q - 2 = 0 \) → \( q = 2 \)

Thus, the values of \( q \) are \( q = 1 \) and \( q = 2 \).

### Step 7: Determine the relationship between \( p \) and \( q \)
Now we have the values:
- \( p = 3 \) or \( p = 4 \)
- \( q = 1 \) or \( q = 2 \)

Since both values of \( p \) (3 and 4) are greater than both values of \( q \) (1 and 2), we can conclude:
\[ p > q \]
or equivalently,
\[ q < p \]

### Final Conclusion
Thus, the relationship between \( p \) and \( q \) is that \( p \) is greater than \( q \).

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