Solve the following quadratic equation and mark and answer as per instructions.
I. ` x^(2) - 7 x - 18 = 0`
II. ` y ^(2) -19 y + 90 = 0`

Options

A. ` x le y`
B. x = y or no relation can be established
C. ` x gt y`
D. ` x ge y `

Answer

To solve the quadratic equations given in the question, we will follow these steps:

### Step 1: Solve the first equation \( x^2 - 7x - 18 = 0 \)

1. **Identify the coefficients**: The equation is in the standard form \( ax^2 + bx + c = 0 \) where \( a = 1 \), \( b = -7 \), and \( c = -18 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( ac = 1 \times -18 = -18 \) and add to \( b = -7 \). The numbers that satisfy this are \( 2 \) and \( -9 \).
3. **Rewrite the equation**: We can express the equation as:
\[ x^2 - 9x + 2x - 18 = 0 \]
4. **Group the terms**: Group the first two and the last two terms:
\[ (x^2 - 9x) + (2x - 18) = 0 \]
5. **Factor by grouping**:
\[ x(x - 9) + 2(x - 9) = 0 \]
This gives:
\[ (x - 9)(x + 2) = 0 \]
6. **Find the roots**: Set each factor to zero:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Step 2: Solve the second equation \( y^2 - 19y + 90 = 0 \)

1. **Identify the coefficients**: The equation is in the form \( ay^2 + by + c = 0 \) where \( a = 1 \), \( b = -19 \), and \( c = 90 \).
2. **Factor the equation**: We need two numbers that multiply to \( ac = 1 \times 90 = 90 \) and add to \( b = -19 \). The numbers that satisfy this are \( -9 \) and \( -10 \).
3. **Rewrite the equation**:
\[ y^2 - 9y - 10y + 90 = 0 \]
4. **Group the terms**:
\[ (y^2 - 9y) + (-10y + 90) = 0 \]
5. **Factor by grouping**:
\[ y(y - 9) - 10(y - 9) = 0 \]
This gives:
\[ (y - 9)(y - 10) = 0 \]
6. **Find the roots**: Set each factor to zero:
\[ y - 9 = 0 \quad \Rightarrow \quad y = 9 \]
\[ y - 10 = 0 \quad \Rightarrow \quad y = 10 \]

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

- From the first equation, we have \( x = 9 \) and \( x = -2 \).
- From the second equation, we have \( y = 9 \) and \( y = 10 \).

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

1. Compare \( x = 9 \) with \( y = 9 \):
- \( x = y \)
2. Compare \( x = 9 \) with \( y = 10 \):
- \( x < y \)
3. Compare \( x = -2 \) with \( y = 9 \):
- \( x < y \)
4. Compare \( x = -2 \) with \( y = 10 \):
- \( x < y \)

### Conclusion

The relationship can be summarized as:
- \( x \leq y \)

### Final Answer

The answer is \( x \leq y \).

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