In each of the following questions, two equations (I) and (II) are given . Solve the equation and mark the correct option :
I. ` x^(2) + 30 x + 224 = 0 `
II. ` y^(2) + 35 y + 306 = 0 `

Options

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

Answer

To solve the given equations and determine the relationship between the values of \(x\) and \(y\), we will follow these steps:

### Step 1: Solve the first equation \(x^2 + 30x + 224 = 0\)

1. **Identify the equation**:
\[ x^2 + 30x + 224 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(224\) (the constant term) and add up to \(30\) (the coefficient of \(x\)). The numbers \(14\) and \(16\) satisfy this condition:
\[ x^2 + 30x + 224 = (x + 14)(x + 16) = 0 \]

3. **Set each factor to zero**:
\[ x + 14 = 0 \quad \text{or} \quad x + 16 = 0 \]

4. **Solve for \(x\)**:
\[ x = -14 \quad \text{or} \quad x = -16 \]

### Step 2: Solve the second equation \(y^2 + 35y + 306 = 0\)

1. **Identify the equation**:
\[ y^2 + 35y + 306 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(306\) and add up to \(35\). The numbers \(17\) and \(18\) satisfy this condition:
\[ y^2 + 35y + 306 = (y + 17)(y + 18) = 0 \]

3. **Set each factor to zero**:
\[ y + 17 = 0 \quad \text{or} \quad y + 18 = 0 \]

4. **Solve for \(y\)**:
\[ y = -17 \quad \text{or} \quad y = -18 \]

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

Now we have the following values:
- For \(x\): \(-14\) and \(-16\)
- For \(y\): \(-17\) and \(-18\)

### Step 4: Determine the relationship between \(x\) and \(y\)

1. **Compare the values**:
- For \(x = -14\):
- Compare with \(y = -17\): \(-14 > -17\)
- Compare with \(y = -18\): \(-14 > -18\)
- For \(x = -16\):
- Compare with \(y = -17\): \(-16 > -17\)
- Compare with \(y = -18\): \(-16 > -18\)

### Conclusion

In both cases, \(x\) is greater than \(y\). Therefore, the relationship we can conclude is:
\[ x > y \]

### Final Answer

The correct option is:
- **Option A: \(x > y\)**