In each of these questions, two equations (I) and (II) are given . You have to solve both the equations and give answer
I. ` x^(2) + 5 x + 6 = 0 `
II. ` y^(2) + 7 y + 12 = 0 `

Options

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

Answer

To solve the given equations step by step, we will start with the first equation and then proceed to the second equation.

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

1. **Factor the quadratic equation**:
We need to factor the equation \( x^2 + 5x + 6 \). We look for two numbers that multiply to \( 6 \) (the constant term) and add up to \( 5 \) (the coefficient of \( x \)).
- The numbers \( 2 \) and \( 3 \) satisfy this condition.

Thus, we can write:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) = 0 \]

2. **Set each factor to zero**:
Now, we set each factor equal to zero:
\[ x + 2 = 0 \quad \text{or} \quad x + 3 = 0 \]

3. **Solve for \( x \)**:
- From \( x + 2 = 0 \), we get \( x = -2 \).
- From \( x + 3 = 0 \), we get \( x = -3 \).

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

### Step 2: Solve the second equation \( y^2 + 7y + 12 = 0 \)

1. **Factor the quadratic equation**:
We need to factor the equation \( y^2 + 7y + 12 \). We look for two numbers that multiply to \( 12 \) and add up to \( 7 \).
- The numbers \( 3 \) and \( 4 \) satisfy this condition.

Thus, we can write:
\[ y^2 + 7y + 12 = (y + 3)(y + 4) = 0 \]

2. **Set each factor to zero**:
Now, we set each factor equal to zero:
\[ y + 3 = 0 \quad \text{or} \quad y + 4 = 0 \]

3. **Solve for \( y \)**:
- From \( y + 3 = 0 \), we get \( y = -3 \).
- From \( y + 4 = 0 \), we get \( y = -4 \).

So, the solutions for the second equation are:
\[ y = -3 \quad \text{and} \quad y = -4 \]

### Step 3: Analyze the results

Now we have the solutions:
- For \( x \): \( -2 \) and \( -3 \)
- For \( y \): \( -3 \) and \( -4 \)

### Step 4: Establish relationships between \( x \) and \( y \)

1. **Compare values**:
- If we take \( x = -2 \) and \( y = -4 \), we find \( x > y \) (since \(-2 > -4\)).
- If we take \( x = -3 \) and \( y = -3 \), we find \( x = y \).
- If we take \( x = -3 \) and \( y = -4 \), we find \( x > y \) (since \(-3 > -4\)).

### Final Conclusion

The relationships we have established are:
- \( x \geq y \)

Thus, the final answer is:
\[ x \geq y \]

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