In each questions two equations numbered I. and II. are given. You have to solve both the equations and mark appropriate answer
I. ` x^(2) + 21 x + 108 = 0 `
II. ` y^(2) + 14 y + 48 = 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 given equations step by step, we will first solve each equation separately and then compare the values of \( x \) and \( y \).

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 21x + 108 = 0 \]

#### Step 1.1: Factor the Quadratic
We need to factor the quadratic equation. We are looking for two numbers that multiply to \( 108 \) (the constant term) and add up to \( 21 \) (the coefficient of \( x \)).

The factors of \( 108 \) that add up to \( 21 \) are \( 12 \) and \( 9 \) because:
\[ 12 \times 9 = 108 \]
\[ 12 + 9 = 21 \]

#### Step 1.2: Rewrite the Equation
We can rewrite the equation as:
\[ x^2 + 12x + 9x + 108 = 0 \]

#### Step 1.3: Group and Factor
Now we group the terms:
\[ (x^2 + 12x) + (9x + 108) = 0 \]
Factoring out common terms gives us:
\[ x(x + 12) + 9(x + 12) = 0 \]
This can be factored further:
\[ (x + 12)(x + 9) = 0 \]

#### Step 1.4: Find Values of \( x \)
Setting each factor to zero gives us:
1. \( x + 12 = 0 \) → \( x = -12 \)
2. \( x + 9 = 0 \) → \( x = -9 \)

### Step 2: Solve Equation II
The second equation is:
\[ y^2 + 14y + 48 = 0 \]

#### Step 2.1: Factor the Quadratic
We need to factor this quadratic as well. We are looking for two numbers that multiply to \( 48 \) and add up to \( 14 \).

The factors of \( 48 \) that add up to \( 14 \) are \( 6 \) and \( 8 \) because:
\[ 6 \times 8 = 48 \]
\[ 6 + 8 = 14 \]

#### Step 2.2: Rewrite the Equation
We can rewrite the equation as:
\[ y^2 + 6y + 8y + 48 = 0 \]

#### Step 2.3: Group and Factor
Now we group the terms:
\[ (y^2 + 6y) + (8y + 48) = 0 \]
Factoring out common terms gives us:
\[ y(y + 6) + 8(y + 6) = 0 \]
This can be factored further:
\[ (y + 6)(y + 8) = 0 \]

#### Step 2.4: Find Values of \( y \)
Setting each factor to zero gives us:
1. \( y + 6 = 0 \) → \( y = -6 \)
2. \( y + 8 = 0 \) → \( y = -8 \)

### Step 3: Compare Values of \( x \) and \( y \)
Now we have the values:
- From Equation I: \( x = -12 \) or \( x = -9 \)
- From Equation II: \( y = -6 \) or \( y = -8 \)

#### Step 3.1: Compare \( x \) and \( y \)
1. For \( x = -12 \):
- \( -12 < -6 \)
- \( -12 < -8 \)

2. For \( x = -9 \):
- \( -9 < -6 \)
- \( -9 < -8 \)

In both cases, we find that \( x < y \).

### Conclusion
The relationship between \( x \) and \( y \) is:
\[ x < y \]

### Final Answer
Thus, the appropriate answer is:
**Option A: \( x < y \)**