I. ` x^(2) + 17x+72 = 0" "` II. ` y^(2) + 19y + 90 = 0`

Options

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

Explanation

To solve the given equations step by step, we will first solve each quadratic equation separately and then compare the values of \(x\) and \(y\).

### Step 1: Solve the first equation \(x^2 + 17x + 72 = 0\)

1. **Identify the coefficients**: Here, \(a = 1\), \(b = 17\), and \(c = 72\).
2. **Factor the quadratic**: We need to find two numbers that multiply to \(72\) (the constant term) and add up to \(17\) (the coefficient of \(x\)).
- The numbers \(8\) and \(9\) satisfy this condition because \(8 \times 9 = 72\) and \(8 + 9 = 17\).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ (x + 8)(x + 9) = 0 \]
4. **Set each factor to zero**:
- \(x + 8 = 0 \implies x = -8\)
- \(x + 9 = 0 \implies x = -9\)

Thus, the solutions for \(x\) are:
\[ x = -8 \quad \text{and} \quad x = -9 \]

### Step 2: Solve the second equation \(y^2 + 19y + 90 = 0\)

1. **Identify the coefficients**: Here, \(a = 1\), \(b = 19\), and \(c = 90\).
2. **Factor the quadratic**: We need to find two numbers that multiply to \(90\) and add up to \(19\).
- The numbers \(9\) and \(10\) satisfy this condition because \(9 \times 10 = 90\) and \(9 + 10 = 19\).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ (y + 9)(y + 10) = 0 \]
4. **Set each factor to zero**:
- \(y + 9 = 0 \implies y = -9\)
- \(y + 10 = 0 \implies y = -10\)

Thus, the solutions for \(y\) are:
\[ y = -9 \quad \text{and} \quad y = -10 \]

### Step 3: Compare the values of \(x\) and \(y\)

Now we have:
- \(x = -8\) or \(x = -9\)
- \(y = -9\) or \(y = -10\)

We can compare the values:
1. If \(x = -8\) and \(y = -9\), then \(x > y\).
2. If \(x = -9\) and \(y = -10\), then \(x > y\).

In both cases, we find that:
\[ x > y \]

### Conclusion

The relation between \(x\) and \(y\) is:
\[ x \geq y \]