I.` x^(2) + 9x + 20 = 0" "II. Y^(2) = 16`

Options

A. if`x gt y`
B. if`x ge y`
C. if `x lt y`
D. if ` x le y`

Explanation

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

### Step 1: Solve the first equation \( x^2 + 9x + 20 = 0 \)

We can factor the quadratic equation. We need two numbers that multiply to \( 20 \) (the constant term) and add up to \( 9 \) (the coefficient of \( x \)). The numbers \( 4 \) and \( 5 \) satisfy this condition.

Thus, we can write:
\[ x^2 + 4x + 5x + 20 = 0 \]
This can be factored as:
\[ (x + 4)(x + 5) = 0 \]

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

Setting each factor to zero gives us:
1. \( x + 4 = 0 \) → \( x = -4 \)
2. \( x + 5 = 0 \) → \( x = -5 \)

So, the solutions for \( x \) are:
\[ x = -4 \quad \text{and} \quad x = -5 \]

### Step 3: Solve the second equation \( y^2 = 16 \)

To find \( y \), we take the square root of both sides:
\[ y = \pm 4 \]
This gives us two possible values for \( y \):
\[ y = 4 \quad \text{and} \quad y = -4 \]

### Step 4: Compare the values of \( x \) and \( y \)

Now we have the values:
- For \( x \): \( -4 \) and \( -5 \)
- For \( y \): \( 4 \) and \( -4 \)

We will compare these values:
1. \( -4 \) (from \( x \)) compared to \( 4 \) (from \( y \)): \( -4 < 4 \)
2. \( -4 \) compared to \( -4 \): \( -4 = -4 \)
3. \( -5 \) compared to \( 4 \): \( -5 < 4 \)
4. \( -5 \) compared to \( -4 \): \( -5 < -4 \)

### Conclusion

From the comparisons:
- \( x \) is less than \( y \) in all cases.
- \( x \) is equal to \( y \) in the case of \( -4 \).

Thus, we can conclude that:
\[ x \leq y \]