I.` 2x^(2)-11x+15 = 0" "II.4y^(2)+13y+9 = 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 \(2x^2 - 11x + 15 = 0\) and \(4y^2 + 13y + 9 = 0\), we will find the values of \(x\) and \(y\) separately and then determine the relationship between them.

### Step 1: Solve the equation \(2x^2 - 11x + 15 = 0\)

1. **Identify the coefficients**:
- \(a = 2\), \(b = -11\), \(c = 15\)

2. **Factor the quadratic equation**:
- We need to find two numbers that multiply to \(a \cdot c = 2 \cdot 15 = 30\) and add to \(b = -11\).
- The numbers are \(-6\) and \(-5\) because \(-6 \cdot -5 = 30\) and \(-6 + -5 = -11\).

3. **Rewrite the equation**:
- \(2x^2 - 6x - 5x + 15 = 0\)

4. **Factor by grouping**:
- Group the terms: \(2x(x - 3) - 5(x - 3) = 0\)
- Factor out \((x - 3)\): \((2x - 5)(x - 3) = 0\)

5. **Set each factor to zero**:
- \(2x - 5 = 0\) or \(x - 3 = 0\)
- Solving these gives:
- \(x = \frac{5}{2}\)
- \(x = 3\)

### Step 2: Solve the equation \(4y^2 + 13y + 9 = 0\)

1. **Identify the coefficients**:
- \(a = 4\), \(b = 13\), \(c = 9\)

2. **Factor the quadratic equation**:
- We need to find two numbers that multiply to \(a \cdot c = 4 \cdot 9 = 36\) and add to \(b = 13\).
- The numbers are \(9\) and \(4\) because \(9 \cdot 4 = 36\) and \(9 + 4 = 13\).

3. **Rewrite the equation**:
- \(4y^2 + 9y + 4y + 9 = 0\)

4. **Factor by grouping**:
- Group the terms: \(y(4y + 9) + 1(4y + 9) = 0\)
- Factor out \((4y + 9)\): \((4y + 9)(y + 1) = 0\)

5. **Set each factor to zero**:
- \(4y + 9 = 0\) or \(y + 1 = 0\)
- Solving these gives:
- \(y = -\frac{9}{4}\)
- \(y = -1\)

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

We have the values:
- \(x = \frac{5}{2} \approx 2.5\) and \(x = 3\)
- \(y = -\frac{9}{4} = -2.25\) and \(y = -1\)

Now, we can compare the values of \(x\) and \(y\):
- For \(x = \frac{5}{2}\) and \(y = -1\): \(x > y\)
- For \(x = 3\) and \(y = -1\): \(x > y\)
- For both values of \(y\), \(x\) is greater.

### Conclusion
The relationship established is that \(x\) is greater than \(y\).