I. `p^(2) - 18 p + 77 = 0 `
II. ` 3 q ^(2) - 25q + 28 = 0`
Quantity I: Value of P
Quantity II: Value of q

Options

A. Quantity I ` gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge `Quantity II
D. Quantity I `le` QuantityII

Answer

To solve the given equations and compare the values of \( p \) and \( q \), we will follow these steps:

### Step 1: Solve the first equation for \( p \)
The first equation is:
\[ p^2 - 18p + 77 = 0 \]

To factor this quadratic equation, we look for two numbers that multiply to \( 77 \) and add up to \( -18 \). The numbers are \( -11 \) and \( -7 \).

Thus, we can rewrite the equation as:
\[ (p - 11)(p - 7) = 0 \]

Setting each factor to zero gives us:
1. \( p - 11 = 0 \) → \( p = 11 \)
2. \( p - 7 = 0 \) → \( p = 7 \)

So, the values of \( p \) are \( 11 \) and \( 7 \).

### Step 2: Solve the second equation for \( q \)
The second equation is:
\[ 3q^2 - 25q + 28 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \( 3 \times 28 = 84 \) and add up to \( -25 \). The numbers are \( -21 \) and \( -4 \).

We can rewrite the equation as:
\[ 3q^2 - 21q - 4q + 28 = 0 \]
Grouping the terms gives us:
\[ 3q(q - 7) - 4(q - 7) = 0 \]

Factoring out \( (q - 7) \) gives:
\[ (3q - 4)(q - 7) = 0 \]

Setting each factor to zero gives us:
1. \( 3q - 4 = 0 \) → \( q = \frac{4}{3} \)
2. \( q - 7 = 0 \) → \( q = 7 \)

So, the values of \( q \) are \( \frac{4}{3} \) and \( 7 \).

### Step 3: Compare the values of \( p \) and \( q \)
Now we have:
- Values of \( p \): \( 11, 7 \)
- Values of \( q \): \( \frac{4}{3}, 7 \)

We will compare these values:
- When \( p = 11 \), \( 11 > 7 \) and \( 11 > \frac{4}{3} \).
- When \( p = 7 \), \( 7 = 7 \) and \( 7 > \frac{4}{3} \).

### Conclusion
From the comparisons:
- The maximum value of \( p \) (which is \( 11 \)) is greater than the maximum value of \( q \) (which is \( 7 \)).
- The value \( 7 \) for \( p \) is equal to the value \( 7 \) for \( q \).

Thus, we conclude that:
**Quantity I (values of \( p \)) is greater than or equal to Quantity II (values of \( q \)).**

### Final Answer
**Quantity I is greater than or equal to Quantity II.**

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