In each of these questions, two equations I. and II. are given . you have to solve both the equations and answer the following questions
I. ` x ^(2) + 13 x + 42 = 0 `
II. ` y^(2) + 8y + 12 = 0 `

Options

A. x = y or no relation
B. ` x lt y`
C. ` x le y `
D. ` x gt y `

Answer

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

### Step 1: Solve the first equation \( I. \: x^2 + 13x + 42 = 0 \)

1. **Identify coefficients**: The equation is in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 13 \), and \( c = 42 \).
2. **Factor the quadratic**: We need to find two numbers that multiply to \( c = 42 \) and add up to \( b = 13 \). The numbers that satisfy this are \( 6 \) and \( 7 \).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ x^2 + 6x + 7x + 42 = 0 \]
4. **Group the terms**: Group the terms to factor by grouping:
\[ (x^2 + 6x) + (7x + 42) = 0 \]
5. **Factor out common terms**:
\[ x(x + 6) + 7(x + 6) = 0 \]
This gives:
\[ (x + 6)(x + 7) = 0 \]
6. **Find the roots**: Set each factor to zero:
\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \]
\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]

### Step 2: Solve the second equation \( II. \: y^2 + 8y + 12 = 0 \)

1. **Identify coefficients**: The equation is in the standard form \( ay^2 + by + c = 0 \), where \( a = 1 \), \( b = 8 \), and \( c = 12 \).
2. **Factor the quadratic**: We need to find two numbers that multiply to \( c = 12 \) and add up to \( b = 8 \). The numbers that satisfy this are \( 6 \) and \( 2 \).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ y^2 + 6y + 2y + 12 = 0 \]
4. **Group the terms**: Group the terms to factor by grouping:
\[ (y^2 + 6y) + (2y + 12) = 0 \]
5. **Factor out common terms**:
\[ y(y + 6) + 2(y + 6) = 0 \]
This gives:
\[ (y + 6)(y + 2) = 0 \]
6. **Find the roots**: Set each factor to zero:
\[ y + 6 = 0 \quad \Rightarrow \quad y = -6 \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]

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

Now we have the values:
- For \( x \): \( -6, -7 \)
- For \( y \): \( -6, -2 \)

1. **Compare \( x = -7 \) with \( y \)**:
- \( -7 < -6 \) (True)
- \( -7 < -2 \) (True)

2. **Compare \( x = -6 \) with \( y \)**:
- \( -6 = -6 \) (True)
- \( -6 < -2 \) (True)

### Conclusion

From the comparisons:
- \( x \) is less than \( y \) when \( x = -7 \).
- \( x \) is equal to \( y \) when \( x = -6 \).
- Therefore, we conclude that:
\[ x \leq y \]

### Final Answer

The relationship between \( x \) and \( y \) is:
\[ x \leq y \]