In the following questions, two equations numbered I and II are given. You have to solve both the equations and
I. `2x^2+19x+42=0`
II. `y^2+9y+20=0`

Options

A. `xgty`
B. `x le y`
C. `x ge y`
D. the relationship cannot be established

Answer

To solve the given equations step by step, we will follow the quadratic formula method and factorization where applicable.

### Step 1: Solve the first equation \(2x^2 + 19x + 42 = 0\)

1. **Identify the coefficients**:
- \(a = 2\), \(b = 19\), \(c = 42\)

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

3. **Calculate the discriminant**:
\[ b^2 - 4ac = 19^2 - 4 \cdot 2 \cdot 42 = 361 - 336 = 25 \]

4. **Find the roots**:
\[ x = \frac{-19 \pm \sqrt{25}}{2 \cdot 2} = \frac{-19 \pm 5}{4} \]
- First root:
\[ x_1 = \frac{-19 + 5}{4} = \frac{-14}{4} = -3.5 \]
- Second root:
\[ x_2 = \frac{-19 - 5}{4} = \frac{-24}{4} = -6 \]

### Step 2: Solve the second equation \(y^2 + 9y + 20 = 0\)

1. **Identify the coefficients**:
- \(a = 1\), \(b = 9\), \(c = 20\)

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

3. **Calculate the discriminant**:
\[ b^2 - 4ac = 9^2 - 4 \cdot 1 \cdot 20 = 81 - 80 = 1 \]

4. **Find the roots**:
\[ y = \frac{-9 \pm \sqrt{1}}{2 \cdot 1} = \frac{-9 \pm 1}{2} \]
- First root:
\[ y_1 = \frac{-9 + 1}{2} = \frac{-8}{2} = -4 \]
- Second root:
\[ y_2 = \frac{-9 - 1}{2} = \frac{-10}{2} = -5 \]

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

- The values obtained are:
- For \(x\): \(x_1 = -3.5\) and \(x_2 = -6\)
- For \(y\): \(y_1 = -4\) and \(y_2 = -5\)

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

- Comparing \(x_1 = -3.5\) with \(y_1 = -4\):
\(-3.5 > -4\)

- Comparing \(x_1 = -3.5\) with \(y_2 = -5\):
\(-3.5 > -5\)

- Comparing \(x_2 = -6\) with \(y_1 = -4\):
\(-6 < -4\)

- Comparing \(x_2 = -6\) with \(y_2 = -5\):
\(-6 < -5\)

### Conclusion

From the comparisons, we see that:
- \(x_1\) is greater than both \(y_1\) and \(y_2\).
- \(x_2\) is less than both \(y_1\) and \(y_2\).

Since there is no consistent relationship established between all values of \(x\) and \(y\), we conclude that the relationship cannot be established.

### Final Answer
The correct option is that the relationship cannot be established.

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