In each of the following questions, two equations (I) and (II) are given . Solve the equation and mark the correct option :
I. ` x^(2) + 23 x+ 132 = 0 `
II. ` y^(2) + 21 y + 110 = 0`

Options

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

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation \( x^2 + 23x + 132 = 0 \)

We will factor the quadratic equation. We need two numbers that multiply to \( 132 \) and add up to \( 23 \).

- The factors of \( 132 \) that add up to \( 23 \) are \( 12 \) and \( 11 \).

Thus, we can rewrite the equation as:
\[ x^2 + 12x + 11x + 132 = 0 \]

Now, we can group the terms:
\[ (x^2 + 12x) + (11x + 132) = 0 \]

Factoring out the common terms, we have:
\[ x(x + 12) + 11(x + 12) = 0 \]

Now, factor out \( (x + 12) \):
\[ (x + 12)(x + 11) = 0 \]

Setting each factor to zero gives us:
\[ x + 12 = 0 \quad \Rightarrow \quad x = -12 \]
\[ x + 11 = 0 \quad \Rightarrow \quad x = -11 \]

So, the values of \( x \) are \( -12 \) and \( -11 \).

### Step 2: Solve the second equation \( y^2 + 21y + 110 = 0 \)

Similarly, we will factor this quadratic equation. We need two numbers that multiply to \( 110 \) and add up to \( 21 \).

- The factors of \( 110 \) that add up to \( 21 \) are \( 11 \) and \( 10 \).

Thus, we can rewrite the equation as:
\[ y^2 + 11y + 10y + 110 = 0 \]

Now, we can group the terms:
\[ (y^2 + 11y) + (10y + 110) = 0 \]

Factoring out the common terms, we have:
\[ y(y + 11) + 10(y + 11) = 0 \]

Now, factor out \( (y + 11) \):
\[ (y + 11)(y + 10) = 0 \]

Setting each factor to zero gives us:
\[ y + 11 = 0 \quad \Rightarrow \quad y = -11 \]
\[ y + 10 = 0 \quad \Rightarrow \quad y = -10 \]

So, the values of \( y \) are \( -11 \) and \( -10 \).

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

We have the values:
- \( x = -12, -11 \)
- \( y = -11, -10 \)

Now, we can compare these values:

1. For \( x = -12 \):
- \( -12 \) is less than \( -11 \) and also less than \( -10 \).

2. For \( x = -11 \):
- \( -11 \) is equal to \( -11 \) and less than \( -10 \).

From these comparisons, we can conclude:
- When \( x = -12 \), \( x < y \) for both values of \( y \).
- When \( x = -11 \), \( x = y \) for \( y = -11 \) and \( x < y \) for \( y = -10 \).

### Final Conclusion

Combining these results, we can say:
\[ x \leq y \]

### Answer

Thus, the correct option is:
**Option D: \( x \leq y \)**

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