In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct options
I. ` x^(2) + 13 x + 42 = 0 `
II. ` y^(2) + 16 y + 63 = 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 first tackle each equation separately and then analyze the results.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 13x + 42 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \(42\) (the constant term) and add up to \(13\) (the coefficient of \(x\)).

The numbers that satisfy these conditions are \(7\) and \(6\).

Thus, we can rewrite the equation as:
\[ (x + 7)(x + 6) = 0 \]

### Step 2: Find the Values of \(x\)
Setting each factor to zero gives us:
1. \( x + 7 = 0 \) → \( x = -7 \)
2. \( x + 6 = 0 \) → \( x = -6 \)

So, the solutions for \(x\) are:
\[ x = -7 \quad \text{and} \quad x = -6 \]

### Step 3: Solve Equation II
The second equation is:
\[ y^2 + 16y + 63 = 0 \]

Similarly, we need to find two numbers that multiply to \(63\) and add up to \(16\).

The numbers that satisfy these conditions are \(9\) and \(7\).

Thus, we can rewrite the equation as:
\[ (y + 9)(y + 7) = 0 \]

### Step 4: Find the Values of \(y\)
Setting each factor to zero gives us:
1. \( y + 9 = 0 \) → \( y = -9 \)
2. \( y + 7 = 0 \) → \( y = -7 \)

So, the solutions for \(y\) are:
\[ y = -9 \quad \text{and} \quad y = -7 \]

### Step 5: Compare the Values of \(x\) and \(y\)
Now we have the values:
- For \(x\): \( -7 \) and \( -6 \)
- For \(y\): \( -9 \) and \( -7 \)

We can compare these values:
1. When \(x = -7\), \(y = -9\) (Here, \(x > y\))
2. When \(x = -6\), \(y = -7\) (Here, \(x > y\))

### Conclusion
In both cases, we find that \(x\) is greater than \(y\). Therefore, we can conclude that:
\[ x \geq y \]

### Final Answer
The correct relation is:
\[ x > y \]