In each of the following questions, two equations (I) and (II) are given. Solve the equations and mark the correct option
I. ` x^(2) + 21 x + 108 = 0 `
II. ` y^(2) + 24 y + 143 = 0`

Options

A. If ` x gt y `
B. If ` x ge y `
C. if ` x lt y `
D. If x = y or no relation can be established between x and y

Answer

To solve the given equations and analyze the relationship between their solutions, we will follow these steps:

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

To factor this quadratic equation, we need to find two numbers that multiply to \( 108 \) (the constant term) and add up to \( 21 \) (the coefficient of \( x \)).

The numbers that satisfy these conditions are \( 12 \) and \( 9 \) because:
- \( 12 \times 9 = 108 \)
- \( 12 + 9 = 21 \)

Now we can rewrite the equation:
\[ x^2 + 12x + 9x + 108 = 0 \]

### Step 2: Factor the Equation
We can group the terms:
\[ (x^2 + 12x) + (9x + 108) = 0 \]
Factoring by grouping:
\[ x(x + 12) + 9(x + 12) = 0 \]
Now factor out the common term:
\[ (x + 12)(x + 9) = 0 \]

### Step 3: Find the Values of \( x \)
Setting each factor to zero gives us:
1. \( x + 12 = 0 \) → \( x = -12 \)
2. \( x + 9 = 0 \) → \( x = -9 \)

So, the solutions for \( x \) are:
\[ x = -12 \quad \text{and} \quad x = -9 \]

### Step 4: Solve Equation II
The second equation is:
\[ y^2 + 24y + 143 = 0 \]

Similarly, we need to find two numbers that multiply to \( 143 \) and add up to \( 24 \). The numbers that fit are \( 13 \) and \( 11 \):
- \( 13 \times 11 = 143 \)
- \( 13 + 11 = 24 \)

Rewriting the equation:
\[ y^2 + 13y + 11y + 143 = 0 \]

### Step 5: Factor the Equation
Grouping the terms:
\[ (y^2 + 13y) + (11y + 143) = 0 \]
Factoring by grouping:
\[ y(y + 13) + 11(y + 13) = 0 \]
Now factor out the common term:
\[ (y + 13)(y + 11) = 0 \]

### Step 6: Find the Values of \( y \)
Setting each factor to zero gives us:
1. \( y + 13 = 0 \) → \( y = -13 \)
2. \( y + 11 = 0 \) → \( y = -11 \)

So, the solutions for \( y \) are:
\[ y = -13 \quad \text{and} \quad y = -11 \]

### Step 7: Analyze the Relationships
Now we have the solutions:
- For \( x \): \( -12 \) and \( -9 \)
- For \( y \): \( -13 \) and \( -11 \)

Now we compare:
1. \( x = -12 \) with \( y = -13 \): \( -12 > -13 \) (so \( x > y \))
2. \( x = -12 \) with \( y = -11 \): \( -12 < -11 \) (so \( x < y \))
3. \( x = -9 \) with \( y = -13 \): \( -9 > -13 \) (so \( x > y \))
4. \( x = -9 \) with \( y = -11 \): \( -9 > -11 \) (so \( x > y \))

### Conclusion
From the comparisons, we see that:
- For \( x = -12 \), it is greater than \( y = -13 \) but less than \( y = -11 \).
- For \( x = -9 \), it is greater than both values of \( y \).

Thus, there is no consistent relationship between \( x \) and \( y \) across all values. Therefore, the answer is that there is **no relation**.