In each of the following questions, two equations (I)and (II) are given. Solve the equations and mark the correct options
I. `x^(2) + 5 x + 6 = 0 `
II. ` y^(2) + 9 y + 14 = 0 `

Options

A. A)If ` x gt y `
B. B)If ` x ge y `
C. C)if ` x lt y `
D. D)If x = y or
no relation can be
established between
x and y

Answer

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

### Step 1: Solve the first equation \( x^2 + 5x + 6 = 0 \)

We will factor the quadratic equation:

1. Identify two numbers that multiply to \( 6 \) (the constant term) and add up to \( 5 \) (the coefficient of \( x \)). The numbers are \( 2 \) and \( 3 \).
2. Rewrite the equation as:
\[ (x + 2)(x + 3) = 0 \]
3. Set each factor to zero:
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

Thus, the solutions for the first equation are:
\[ x = -2 \quad \text{and} \quad x = -3 \]

### Step 2: Solve the second equation \( y^2 + 9y + 14 = 0 \)

Similarly, we will factor this quadratic equation:

1. Identify two numbers that multiply to \( 14 \) (the constant term) and add up to \( 9 \) (the coefficient of \( y \)). The numbers are \( 7 \) and \( 2 \).
2. Rewrite the equation as:
\[ (y + 7)(y + 2) = 0 \]
3. Set each factor to zero:
\[ y + 7 = 0 \quad \Rightarrow \quad y = -7 \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]

Thus, the solutions for the second equation are:
\[ y = -7 \quad \text{and} \quad y = -2 \]

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

Now we have the following values:
- From the first equation: \( x = -2 \) and \( x = -3 \)
- From the second equation: \( y = -7 \) and \( y = -2 \)

We will compare these values:
1. For \( x = -2 \) and \( y = -2 \):
\[ x = y \]
2. For \( x = -3 \) and \( y = -7 \):
\[ x > y \quad \text{(since -3 is greater than -7)} \]

### Conclusion

From the comparisons, we can conclude:
- \( x \) can be equal to \( y \) when both are \( -2 \).
- \( x \) is greater than \( y \) when \( x = -3 \) and \( y = -7 \).

Thus, the final relationship we can establish is:
\[ x \geq y \]