In each of the following questions, two equations (I) and (II) are given. Solve the equations and mark the correct option
I. `x^(2) + 23 x + 130 = 0 `
II. ` y^(2) + 30 y + 224 = 0 `

Options

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

Answer

To solve the given equations step by step, we will follow these steps:

### Step 1: Solve the first equation \( x^2 + 23x + 130 = 0 \)

1. **Identify the equation**: The equation is \( x^2 + 23x + 130 = 0 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( 130 \) (the constant term) and add up to \( 23 \) (the coefficient of \( x \)).
- The numbers \( 13 \) and \( 10 \) satisfy this condition because \( 13 + 10 = 23 \) and \( 13 \times 10 = 130 \).
3. **Rewrite the equation**: We can factor the quadratic as:
\[ (x + 13)(x + 10) = 0 \]
4. **Set each factor to zero**:
- \( x + 13 = 0 \) gives \( x = -13 \)
- \( x + 10 = 0 \) gives \( x = -10 \)

### Step 2: Solve the second equation \( y^2 + 30y + 224 = 0 \)

1. **Identify the equation**: The equation is \( y^2 + 30y + 224 = 0 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( 224 \) and add up to \( 30 \).
- The numbers \( 16 \) and \( 14 \) satisfy this condition because \( 16 + 14 = 30 \) and \( 16 \times 14 = 224 \).
3. **Rewrite the equation**: We can factor the quadratic as:
\[ (y + 16)(y + 14) = 0 \]
4. **Set each factor to zero**:
- \( y + 16 = 0 \) gives \( y = -16 \)
- \( y + 14 = 0 \) gives \( y = -14 \)

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

1. **List the values obtained**:
- From the first equation: \( x = -13, -10 \)
- From the second equation: \( y = -16, -14 \)
2. **Compare the values**:
- For \( x = -13 \):
- Compare with \( y = -16 \): \( -13 > -16 \) (True)
- Compare with \( y = -14 \): \( -13 > -14 \) (True)
- For \( x = -10 \):
- Compare with \( y = -16 \): \( -10 > -16 \) (True)
- Compare with \( y = -14 \): \( -10 > -14 \) (True)

### Conclusion

In all cases, \( x \) is greater than \( y \). Therefore, the final relation is:
\[ x > y \]

### Final Answer

The correct option is \( x > y \).

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