In each question two equations numbered I and II are given. You have to solve both the equations and mark the answer
I. `4x^2 - 15x + 14=0`
II. `6y^2-10y+4=0`

Options

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

Answer

To solve the given equations step by step, we will first tackle each quadratic equation individually.

### Step 1: Solve the first equation \(4x^2 - 15x + 14 = 0\)

1. **Identify coefficients**: The equation is in the standard form \(ax^2 + bx + c = 0\), where \(a = 4\), \(b = -15\), and \(c = 14\).

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**:
\[ b^2 - 4ac = (-15)^2 - 4 \cdot 4 \cdot 14 = 225 - 224 = 1 \]

4. **Substitute into the formula**:
\[ x = \frac{-(-15) \pm \sqrt{1}}{2 \cdot 4} = \frac{15 \pm 1}{8} \]

5. **Calculate the two possible values for \(x\)**:
- \(x_1 = \frac{15 + 1}{8} = \frac{16}{8} = 2\)
- \(x_2 = \frac{15 - 1}{8} = \frac{14}{8} = \frac{7}{4}\)

Thus, the solutions for the first equation are \(x = 2\) and \(x = \frac{7}{4}\).

### Step 2: Solve the second equation \(6y^2 - 10y + 4 = 0\)

1. **Identify coefficients**: The equation is in the standard form \(ay^2 + by + c = 0\), where \(a = 6\), \(b = -10\), and \(c = 4\).

2. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-10)^2 - 4 \cdot 6 \cdot 4 = 100 - 96 = 4 \]

4. **Substitute into the formula**:
\[ y = \frac{-(-10) \pm \sqrt{4}}{2 \cdot 6} = \frac{10 \pm 2}{12} \]

5. **Calculate the two possible values for \(y\)**:
- \(y_1 = \frac{10 + 2}{12} = \frac{12}{12} = 1\)
- \(y_2 = \frac{10 - 2}{12} = \frac{8}{12} = \frac{2}{3}\)

Thus, the solutions for the second equation are \(y = 1\) and \(y = \frac{2}{3}\).

### Step 3: Compare the values of \(x\) and \(y\)

Now we have the values:
- From the first equation: \(x = 2\) and \(x = \frac{7}{4}\)
- From the second equation: \(y = 1\) and \(y = \frac{2}{3}\)

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

- The larger value of \(x\) is \(2\), and the larger value of \(y\) is \(1\).
- Therefore, \(2 > 1\) and \(\frac{7}{4} > \frac{2}{3}\).

### Conclusion

From the comparisons, we can conclude that \(x\) is greater than \(y\).

### Final Answer

The correct relation is \(x > y\).

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