I. ` x^(2) - 72 = x" "` II. ` y^(2) = 64`

Options

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

Explanation

To solve the equations step by step, we will break down each equation and find the values of \( x \) and \( y \).

### Step 1: Solve the first equation \( x^2 - 72 = x \)

1. Rearrange the equation:
\[ x^2 - x - 72 = 0 \]

### Step 2: Factor the quadratic equation

2. We need to factor the quadratic equation \( x^2 - x - 72 = 0 \). We look for two numbers that multiply to \(-72\) and add to \(-1\). The numbers are \( -9 \) and \( 8 \).
\[ (x - 9)(x + 8) = 0 \]

### Step 3: Find the values of \( x \)

3. Set each factor to zero:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \]
Thus, the solutions for \( x \) are \( x = 9 \) and \( x = -8 \).

### Step 4: Solve the second equation \( y^2 = 64 \)

4. Take the square root of both sides:
\[ y = \pm 8 \]
Thus, the solutions for \( y \) are \( y = 8 \) and \( y = -8 \).

### Step 5: Compare the values of \( x \) and \( y \)

5. We have the values:
- For \( x \): \( 9, -8 \)
- For \( y \): \( 8, -8 \)

6. Now we compare the values:
- The maximum value of \( x \) is \( 9 \) which is greater than the maximum value of \( y \) which is \( 8 \).
- The minimum value of \( x \) is \( -8 \) which is equal to the minimum value of \( y \) which is also \( -8 \).

### Conclusion

7. The relation between \( x \) and \( y \) can be summarized as:
\[ x \geq y \]