(i)` x^(2) -18x+72 = 0`
(ii)` y^(2)=36`

Options

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

Explanation

Let's solve the equations step by step.

### Part (i): Solve the equation \( x^2 - 18x + 72 = 0 \)

1. **Identify the coefficients**: In the quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = -18 \)
- \( c = 72 \)

2. **Use the quadratic formula**: The solutions for \( x \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Calculate the discriminant**: First, we calculate \( b^2 - 4ac \):
\[ b^2 - 4ac = (-18)^2 - 4 \cdot 1 \cdot 72 = 324 - 288 = 36 \]

4. **Substitute into the quadratic formula**:
\[ x = \frac{-(-18) \pm \sqrt{36}}{2 \cdot 1} = \frac{18 \pm 6}{2} \]

5. **Calculate the two possible values for \( x \)**:
- First solution:
\[ x_1 = \frac{18 + 6}{2} = \frac{24}{2} = 12 \]
- Second solution:
\[ x_2 = \frac{18 - 6}{2} = \frac{12}{2} = 6 \]

6. **Final solutions**: The solutions to the equation \( x^2 - 18x + 72 = 0 \) are:
\[ x = 12 \quad \text{and} \quad x = 6 \]

### Part (ii): Solve the equation \( y^2 = 36 \)

1. **Take the square root of both sides**:
\[ y = \pm \sqrt{36} \]

2. **Calculate the square root**:
\[ y = \pm 6 \]

3. **Final solutions**: The solutions to the equation \( y^2 = 36 \) are:
\[ y = 6 \quad \text{and} \quad y = -6 \]

### Summary of Solutions:
- For \( x^2 - 18x + 72 = 0 \): \( x = 12 \) and \( x = 6 \)
- For \( y^2 = 36 \): \( y = 6 \) and \( y = -6 \)