Solve the given quadratic equations and mark the correct option based on your answer -
I. ` x^(2) = 81 `
II. ` y^(2) - 18 y + 81 = 0`

Options

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

Answer

To solve the given quadratic equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation \( x^2 = 81 \)

1. Rewrite the equation:
\[ x^2 - 81 = 0 \]

2. Factor the equation using the difference of squares:
\[ (x + 9)(x - 9) = 0 \]

3. Set each factor to zero:
- \( x + 9 = 0 \) gives \( x = -9 \)
- \( x - 9 = 0 \) gives \( x = 9 \)

Thus, the solutions for \( x \) are:
\[ x = -9 \quad \text{and} \quad x = 9 \]

### Step 2: Solve the second equation \( y^2 - 18y + 81 = 0 \)

1. Recognize that this is a perfect square trinomial:
\[ y^2 - 2 \cdot 9 \cdot y + 9^2 = 0 \]

2. Factor the equation:
\[ (y - 9)(y - 9) = 0 \]

3. Set the factor to zero:
- \( y - 9 = 0 \) gives \( y = 9 \)

Thus, the solution for \( y \) is:
\[ y = 9 \]

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

We have:
- From the first equation: \( x = -9 \) or \( x = 9 \)
- From the second equation: \( y = 9 \)

Now we compare the values:
1. When \( x = -9 \):
- \( -9 < 9 \) (so \( x < y \))

2. When \( x = 9 \):
- \( 9 = 9 \) (so \( x = y \))

### Conclusion

Combining the results, we find:
\[ x \leq y \]

### Final Answer

The correct option based on our analysis is:
\[ \text{Option D: } x \leq y \]