I. ` 3x^(2) - 7x+2 = 0" " `
II. ` 2y^(2) - 11y + 15 = 0`

Options

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

Explanation

To solve the equations step by step, we will start with the first equation and then move on to the second equation.

### Step 1: Solve the first equation \(3x^2 - 7x + 2 = 0\)

1. **Identify the coefficients**: Here, \(a = 3\), \(b = -7\), and \(c = 2\).
2. **Use the quadratic formula**: The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot 2 = 49 - 24 = 25 \]
4. **Substitute into the quadratic formula**:
\[ x = \frac{-(-7) \pm \sqrt{25}}{2 \cdot 3} = \frac{7 \pm 5}{6} \]
5. **Calculate the two possible values for \(x\)**:
- First value:
\[ x_1 = \frac{7 + 5}{6} = \frac{12}{6} = 2 \]
- Second value:
\[ x_2 = \frac{7 - 5}{6} = \frac{2}{6} = \frac{1}{3} \]

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

1. **Identify the coefficients**: Here, \(a = 2\), \(b = -11\), and \(c = 15\).
2. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-11)^2 - 4 \cdot 2 \cdot 15 = 121 - 120 = 1 \]
4. **Substitute into the quadratic formula**:
\[ y = \frac{-(-11) \pm \sqrt{1}}{2 \cdot 2} = \frac{11 \pm 1}{4} \]
5. **Calculate the two possible values for \(y\)**:
- First value:
\[ y_1 = \frac{11 + 1}{4} = \frac{12}{4} = 3 \]
- Second value:
\[ y_2 = \frac{11 - 1}{4} = \frac{10}{4} = \frac{5}{2} \]

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

- The values obtained are:
- For \(x\): \(2\) and \(\frac{1}{3}\)
- For \(y\): \(3\) and \(\frac{5}{2}\)

Now we can compare these values:
- \(x = 2\) and \(y = 3\) → \(x < y\)
- \(x = \frac{1}{3}\) and \(y = 3\) → \(x < y\)
- \(x = 2\) and \(y = \frac{5}{2}\) → \(x > y\) (not applicable since we need to compare all values)

Thus, in both cases, we find that \(x\) can be less than \(y\).

### Final Conclusion:
The relationship is that \(x\) can be less than \(y\).

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