In each question two equations numbered I and II are given. You have to solve both the equations and mark the answer
I. `x^2+11x+18=0`
II.`y^2 -sqrt81 =0`

Options

A. `x gt y`
B. `x ge y `
C. `x lt y `
D. 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 \( I: x^2 + 11x + 18 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 11 \), and \( c = 18 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \( c \) (18) and add up to \( b \) (11). The numbers are 9 and 2.
3. **Rewrite the equation**:
\[ x^2 + 9x + 2x + 18 = 0 \]
4. **Group the terms**:
\[ (x^2 + 9x) + (2x + 18) = 0 \]
5. **Factor by grouping**:
\[ x(x + 9) + 2(x + 9) = 0 \]
\[ (x + 9)(x + 2) = 0 \]
6. **Set each factor to zero**:
\[ x + 9 = 0 \quad \text{or} \quad x + 2 = 0 \]
7. **Solve for \( x \)**:
\[ x = -9 \quad \text{or} \quad x = -2 \]

### Step 2: Solve the second equation \( II: y^2 - \sqrt{81} = 0 \)

1. **Calculate \( \sqrt{81} \)**:
\[ \sqrt{81} = 9 \]
2. **Rewrite the equation**:
\[ y^2 - 9 = 0 \]
3. **Add 9 to both sides**:
\[ y^2 = 9 \]
4. **Take the square root of both sides**:
\[ y = \pm 3 \]
This gives us two values for \( y \):
\[ y = 3 \quad \text{or} \quad y = -3 \]

### Step 3: Analyze the relationship between \( x \) and \( y \)

1. **Values obtained**:
- From equation I, \( x = -9 \) or \( x = -2 \).
- From equation II, \( y = 3 \) or \( y = -3 \).
2. **Compare the values**:
- \( y = 3 \) is greater than both \( x = -9 \) and \( x = -2 \).
- \( y = -3 \) is greater than \( x = -9 \) but less than \( x = -2 \).

### Conclusion

Since there is no consistent relationship established between the values of \( x \) and \( y \) (i.e., one is not consistently greater than or less than the other), we conclude that:

**The correct answer is: No relationship can be established between \( x \) and \( y \).**

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