Solve the equations given below and give answer
1) if `x lt y` 2) if `x gt y`
3) if `c ge y` 4) if `x le y`
5) if x=y or no relation can be established
I. `2x^2-(4+sqrt13)x+2sqrt13=0` II. `10y^2-(18+5sqrt13)y+9sqrt13=0`

Options

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

Answer

To solve the equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation for \( x \)
The first equation is:
\[ 2x^2 - (4 + \sqrt{13})x + 2\sqrt{13} = 0 \]
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -(4 + \sqrt{13}) \), and \( c = 2\sqrt{13} \).

1. Calculate \( b^2 - 4ac \):
\[ b^2 = (4 + \sqrt{13})^2 = 16 + 8\sqrt{13} + 13 = 29 + 8\sqrt{13} \]
\[ 4ac = 4 \cdot 2 \cdot 2\sqrt{13} = 16\sqrt{13} \]
\[ b^2 - 4ac = (29 + 8\sqrt{13}) - 16\sqrt{13} = 29 - 8\sqrt{13} \]

2. Now substitute back into the quadratic formula:
\[ x = \frac{4 + \sqrt{13} \pm \sqrt{29 - 8\sqrt{13}}}{4} \]

### Step 2: Solve the second equation for \( y \)
The second equation is:
\[ 10y^2 - (18 + 5\sqrt{13})y + 9\sqrt{13} = 0 \]
Using the quadratic formula again with \( a = 10 \), \( b = -(18 + 5\sqrt{13}) \), and \( c = 9\sqrt{13} \):

1. Calculate \( b^2 - 4ac \):
\[ b^2 = (18 + 5\sqrt{13})^2 = 324 + 180\sqrt{13} + 25 \cdot 13 = 324 + 180\sqrt{13} + 325 = 649 + 180\sqrt{13} \]
\[ 4ac = 4 \cdot 10 \cdot 9\sqrt{13} = 360\sqrt{13} \]
\[ b^2 - 4ac = (649 + 180\sqrt{13}) - 360\sqrt{13} = 649 - 180\sqrt{13} \]

2. Substitute back into the quadratic formula:
\[ y = \frac{18 + 5\sqrt{13} \pm \sqrt{649 - 180\sqrt{13}}}{20} \]

### Step 3: Compare values of \( x \) and \( y \)
Now we have two values for \( x \) and two values for \( y \) from the quadratic equations. We need to analyze the relationships:

1. **If \( x < y \)**: Compare the values of \( x \) and \( y \) obtained from the equations.
2. **If \( x > y \)**: Similarly, check if \( x \) is greater than \( y \).
3. **If \( c \geq y \)**: Check if the calculated \( c \) value is greater than or equal to \( y \).
4. **If \( x \leq y \)**: Again, check if \( x \) is less than or equal to \( y \).
5. **If \( x = y \)**: Finally, check if both values are equal.

### Conclusion
After performing the calculations and comparisons, we conclude that:
- If \( x = 2 \) and \( y = \frac{9}{5} \), then \( x > y \).
- If \( x = \frac{\sqrt{13}}{2} \) and \( y = \frac{9}{5} \), then \( x < y \).

Thus, the relationships can be summarized as:
- \( x > y \) in some cases and \( x < y \) in others.