I.` a^(2) - 11a + 24 = 0`
II.` b^(2) + 3b - 18 = 0`

Options

A. if `a gt b`
B. if` a ge b`
C. if a = b or no relation can be established between a and b.
D. if` a le b`

Explanation

To solve the equations and find the relationship between \( a \) and \( b \), we will follow these steps:

### Step 1: Solve the first equation \( a^2 - 11a + 24 = 0 \)

We will factor the quadratic equation:

1. Identify two numbers that multiply to \( 24 \) (the constant term) and add up to \( -11 \) (the coefficient of \( a \)).
2. The numbers \( -8 \) and \( -3 \) satisfy this condition since \( -8 \times -3 = 24 \) and \( -8 + -3 = -11 \).
3. Thus, we can factor the equation as:
\[ (a - 8)(a - 3) = 0 \]

### Step 2: Find the values of \( a \)

Setting each factor to zero gives us:
1. \( a - 8 = 0 \) → \( a = 8 \)
2. \( a - 3 = 0 \) → \( a = 3 \)

So, the values of \( a \) are \( 3 \) and \( 8 \).

### Step 3: Solve the second equation \( b^2 + 3b - 18 = 0 \)

We will factor this quadratic equation as well:

1. Identify two numbers that multiply to \( -18 \) (the constant term) and add up to \( 3 \) (the coefficient of \( b \)).
2. The numbers \( 6 \) and \( -3 \) satisfy this condition since \( 6 \times -3 = -18 \) and \( 6 + (-3) = 3 \).
3. Thus, we can factor the equation as:
\[ (b + 6)(b - 3) = 0 \]

### Step 4: Find the values of \( b \)

Setting each factor to zero gives us:
1. \( b + 6 = 0 \) → \( b = -6 \)
2. \( b - 3 = 0 \) → \( b = 3 \)

So, the values of \( b \) are \( -6 \) and \( 3 \).

### Step 5: Determine the relationship between \( a \) and \( b \)

Now, we will analyze the values we found:
- Values of \( a \): \( 3, 8 \)
- Values of \( b \): \( -6, 3 \)

From the values:
- The maximum value of \( b \) is \( 3 \), which is equal to the minimum value of \( a \) (also \( 3 \)).
- The other value of \( b \) is \( -6 \), which is less than both values of \( a \).

Thus, we can conclude:
\[ b \leq a \]
or equivalently,
\[ a \geq b \]

### Final Conclusion

The relationship between \( a \) and \( b \) is that \( a \) is greater than or equal to \( b \).

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