Two equations (I) and (II) are given in each question. On the basis of these equations you have to decide the relation between x and y and give answer
I. ` x^(2) + 5 x + 6 = 0 `
II. ` y^(2) + 3y + 2 = 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 and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 5x + 6 = 0 \]

We can factor this quadratic equation. We need two numbers that multiply to \( 6 \) (the constant term) and add up to \( 5 \) (the coefficient of \( x \)). The numbers \( 2 \) and \( 3 \) satisfy these conditions.

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

Setting each factor to zero gives us:
1. \( x + 2 = 0 \) → \( x = -2 \)
2. \( x + 3 = 0 \) → \( x = -3 \)

So, the solutions for \( x \) are:
\[ x = -2 \quad \text{or} \quad x = -3 \]

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

Similarly, we will factor this equation. We need two numbers that multiply to \( 2 \) and add up to \( 3 \). The numbers \( 1 \) and \( 2 \) work here.

Thus, we can write:
\[ (y + 1)(y + 2) = 0 \]

Setting each factor to zero gives us:
1. \( y + 1 = 0 \) → \( y = -1 \)
2. \( y + 2 = 0 \) → \( y = -2 \)

So, the solutions for \( y \) are:
\[ y = -1 \quad \text{or} \quad y = -2 \]

### Step 3: Compare Values of \( x \) and \( y \)
Now we have:
- Possible values for \( x \): \( -2 \) and \( -3 \)
- Possible values for \( y \): \( -1 \) and \( -2 \)

We will compare these values:
1. If \( x = -2 \):
- \( y = -1 \) → \( -2 < -1 \)
- \( y = -2 \) → \( -2 = -2 \)

2. If \( x = -3 \):
- \( y = -1 \) → \( -3 < -1 \)
- \( y = -2 \) → \( -3 < -2 \)

### Conclusion
From the comparisons:
- In both cases, \( x \) is less than or equal to \( y \).

Thus, the relationship can be summarized as:
\[ x \leq y \]

### Final Answer
The correct answer is:
\[ x \leq y \]

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