I.` x^(2) - 36x+324 = 0" "II.3y^(2)+17y+24 = 0`

Options

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

Explanation

To solve the equations step by step, we will tackle each equation separately and then find the relationship between the values of \( x \) and \( y \).

### Step 1: Solve the first equation \( x^2 - 36x + 324 = 0 \)

1. **Identify the equation**: We have \( x^2 - 36x + 324 = 0 \).
2. **Recognize it as a perfect square**: This can be rewritten as \( (x - 18)^2 = 0 \).
3. **Set the expression to zero**: Thus, \( x - 18 = 0 \).
4. **Solve for \( x \)**: Therefore, \( x = 18 \).

### Step 2: Solve the second equation \( 3y^2 + 17y + 24 = 0 \)

1. **Identify the equation**: We have \( 3y^2 + 17y + 24 = 0 \).
2. **Factor the quadratic**: We need to find two numbers that multiply to \( 3 \times 24 = 72 \) and add to \( 17 \). The numbers \( 9 \) and \( 8 \) work since \( 9 + 8 = 17 \).
3. **Rewrite the middle term**: We can rewrite the equation as \( 3y^2 + 9y + 8y + 24 = 0 \).
4. **Group the terms**: This can be grouped as \( (3y^2 + 9y) + (8y + 24) = 0 \).
5. **Factor by grouping**: Factor out common terms: \( 3y(y + 3) + 8(y + 3) = 0 \).
6. **Factor out the common binomial**: This gives us \( (3y + 8)(y + 3) = 0 \).
7. **Set each factor to zero**:
- For \( 3y + 8 = 0 \), we get \( y = -\frac{8}{3} \).
- For \( y + 3 = 0 \), we get \( y = -3 \).

### Step 3: Summary of Solutions

- The solutions for \( x \) is \( x = 18 \).
- The solutions for \( y \) are \( y = -3 \) and \( y = -\frac{8}{3} \).

### Step 4: Find the relationship between \( x \) and \( y \)

1. **Compare values**:
- For \( y = -3 \): \( x = 18 > -3 \).
- For \( y = -\frac{8}{3} \): \( x = 18 > -\frac{8}{3} \).
2. **Conclusion**: In both cases, \( x \) is greater than \( y \).

### Final Result

Thus, we conclude that \( x > y \).

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