In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct options :
I. ` x^(2) + 18 x + 77 = 0 `
II. ` y^(2) + 22y + 117 = 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 determine the relationship between the variables \(x\) and \(y\), we will follow these steps:

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 18x + 77 = 0 \]

To factor this equation, we need two numbers that multiply to \(77\) and add up to \(18\). The numbers \(11\) and \(7\) satisfy this condition:
- \(11 + 7 = 18\)
- \(11 \times 7 = 77\)

Thus, we can rewrite the equation as:
\[ (x + 11)(x + 7) = 0 \]

Setting each factor to zero gives:
1. \(x + 11 = 0 \Rightarrow x = -11\)
2. \(x + 7 = 0 \Rightarrow x = -7\)

### Step 2: Solve Equation II
The second equation is:
\[ y^2 + 22y + 117 = 0 \]

Similarly, we need to find two numbers that multiply to \(117\) and add up to \(22\). The numbers \(13\) and \(9\) satisfy this condition:
- \(13 + 9 = 22\)
- \(13 \times 9 = 117\)

Thus, we can rewrite the equation as:
\[ (y + 13)(y + 9) = 0 \]

Setting each factor to zero gives:
1. \(y + 13 = 0 \Rightarrow y = -13\)
2. \(y + 9 = 0 \Rightarrow y = -9\)

### Step 3: Compare Values of \(x\) and \(y\)
Now we have the values:
- From Equation I: \(x = -11\) and \(x = -7\)
- From Equation II: \(y = -13\) and \(y = -9\)

We will compare these values to determine the relationships:
1. For \(x = -7\):
- Compare with \(y = -9\): \(-7 > -9\)
- Compare with \(y = -13\): \(-7 > -13\)

2. For \(x = -11\):
- Compare with \(y = -9\): \(-11 < -9\)
- Compare with \(y = -13\): \(-11 > -13\)

### Conclusion
From the comparisons:
- When \(x = -7\), \(x\) is greater than both \(y = -9\) and \(y = -13\).
- When \(x = -11\), \(x\) is less than \(y = -9\) but greater than \(y = -13\).

Since we have both comparisons yielding different results, we conclude that there is no consistent relationship between \(x\) and \(y\).

### Final Answer
Thus, the correct option is that no relation can be established between \(x\) and \(y\).

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