Each of the following questions is provided with 2 statements i.e. Statement I & Statement II. You have the solve them and mark the correct option
I. ` x^(2) + 8 x + 12 = 0 `
II. ` 2 y^(2) + 14 y + 24 = 0 `

Options

A. ` x gt y`
B. ` x lt y `
C. ` x le y `
D. x = y or no relation can be established

Answer

To solve the given problem, we will analyze both statements separately and find the values of \( x \) and \( y \). Then, we will compare these values to determine the relationship between \( x \) and \( y \).

### Step 1: Solve Statement I

The equation given in Statement I is:
\[ x^2 + 8x + 12 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \( 12 \) (the constant term) and add up to \( 8 \) (the coefficient of \( x \)). The numbers \( 6 \) and \( 2 \) satisfy these conditions.

Thus, we can factor the equation as:
\[ (x + 2)(x + 6) = 0 \]

Setting each factor to zero gives us the solutions for \( x \):
1. \( x + 2 = 0 \) → \( x = -2 \)
2. \( x + 6 = 0 \) → \( x = -6 \)

So, the possible values for \( x \) are \( -2 \) and \( -6 \).

### Step 2: Solve Statement II

The equation given in Statement II is:
\[ 2y^2 + 14y + 24 = 0 \]

First, we can simplify this equation by dividing all terms by \( 2 \):
\[ y^2 + 7y + 12 = 0 \]

Next, we need to find two numbers that multiply to \( 12 \) and add up to \( 7 \). The numbers \( 3 \) and \( 4 \) meet these criteria.

Thus, we can factor the equation as:
\[ (y + 3)(y + 4) = 0 \]

Setting each factor to zero gives us the solutions for \( y \):
1. \( y + 3 = 0 \) → \( y = -3 \)
2. \( y + 4 = 0 \) → \( y = -4 \)

So, the possible values for \( y \) are \( -3 \) and \( -4 \).

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

Now we have the values:
- For \( x \): \( -2 \) and \( -6 \)
- For \( y \): \( -3 \) and \( -4 \)

We will compare these values:

1. **First comparison**:
- \( x = -2 \) and \( y = -3 \):
\[ -2 > -3 \quad \text{(True)} \]

2. **Second comparison**:
- \( x = -2 \) and \( y = -4 \):
\[ -2 > -4 \quad \text{(True)} \]

3. **Third comparison**:
- \( x = -6 \) and \( y = -3 \):
\[ -6 < -3 \quad \text{(True)} \]

4. **Fourth comparison**:
- \( x = -6 \) and \( y = -4 \):
\[ -6 < -4 \quad \text{(True)} \]

### Conclusion

From the comparisons, we see that:
- When \( x = -2 \), \( x \) is greater than both values of \( y \).
- When \( x = -6 \), \( x \) is less than both values of \( y \).

Since \( x \) can be both greater than and less than \( y \) depending on the values chosen, we conclude that there is **no consistent relationship** between \( x \) and \( y \).

### Final Answer
No relation can be established between \( x \) and \( y \).

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