In the following questions, there are two equations in x and y . You have to solve both the equations and give answer
I. ` x^(2) + 21 x + 110 = 0`
II. ` y ^(2) + 17 y + 72 = 0`

Options

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

Answer

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

### Step 1: Solve the first equation \( x^2 + 21x + 110 = 0 \)

1. **Identify the coefficients**: The equation is in the standard quadratic form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = 21 \)
- \( c = 110 \)

2. **Find the factors of \( c \) (110)**: We need two numbers that multiply to \( 110 \) and add up to \( 21 \). The factors of \( 110 \) that satisfy this condition are \( 11 \) and \( 10 \).

3. **Rewrite the equation**: We can express the middle term using the factors:
\[ x^2 + 11x + 10x + 110 = 0 \]

4. **Factor by grouping**:
\[ (x^2 + 11x) + (10x + 110) = 0 \]
\[ x(x + 11) + 10(x + 11) = 0 \]
\[ (x + 11)(x + 10) = 0 \]

5. **Set each factor to zero**:
\[ x + 11 = 0 \quad \Rightarrow \quad x = -11 \]
\[ x + 10 = 0 \quad \Rightarrow \quad x = -10 \]

### Step 2: Solve the second equation \( y^2 + 17y + 72 = 0 \)

1. **Identify the coefficients**: The equation is in the standard quadratic form \( ay^2 + by + c = 0 \), where:
- \( a = 1 \)
- \( b = 17 \)
- \( c = 72 \)

2. **Find the factors of \( c \) (72)**: We need two numbers that multiply to \( 72 \) and add up to \( 17 \). The factors of \( 72 \) that satisfy this condition are \( 8 \) and \( 9 \).

3. **Rewrite the equation**: We can express the middle term using the factors:
\[ y^2 + 8y + 9y + 72 = 0 \]

4. **Factor by grouping**:
\[ (y^2 + 8y) + (9y + 72) = 0 \]
\[ y(y + 8) + 9(y + 8) = 0 \]
\[ (y + 8)(y + 9) = 0 \]

5. **Set each factor to zero**:
\[ y + 8 = 0 \quad \Rightarrow \quad y = -8 \]
\[ y + 9 = 0 \quad \Rightarrow \quad y = -9 \]

### Step 3: Compare the values of x and y

Now we have the values:
- From the first equation: \( x = -11, -10 \)
- From the second equation: \( y = -8, -9 \)

We will compare these values:

1. Compare \( x = -11 \) and \( y = -9 \):
\[ -11 < -9 \quad \text{(True)} \]

2. Compare \( x = -10 \) and \( y = -9 \):
\[ -10 < -9 \quad \text{(True)} \]

3. Compare \( x = -11 \) and \( y = -8 \):
\[ -11 < -8 \quad \text{(True)} \]

4. Compare \( x = -10 \) and \( y = -8 \):
\[ -10 < -8 \quad \text{(True)} \]

### Conclusion:
From all comparisons, we can conclude that \( x < y \).