Solve the given quadratic equations and mark the correct options based on your answer
I. ` x^(2) + x - 12 = 0`
II. ` y^(2) + 2y - 15 = 0 `

Options

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

Answer

To solve the given quadratic equations and find the relationship between the solutions, we will proceed step by step.

### Step 1: Solve the first quadratic equation \( x^2 + x - 12 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 1 \), and \( c = -12 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( -12 \) (the constant term) and add up to \( 1 \) (the coefficient of \( x \)).
- The numbers \( 4 \) and \( -3 \) satisfy this condition because \( 4 \times (-3) = -12 \) and \( 4 + (-3) = 1 \).
3. **Write the factorization**:
\[ (x + 4)(x - 3) = 0 \]
4. **Set each factor to zero**:
- \( x + 4 = 0 \) gives \( x = -4 \)
- \( x - 3 = 0 \) gives \( x = 3 \)

### Step 2: Solve the second quadratic equation \( y^2 + 2y - 15 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 2 \), and \( c = -15 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( -15 \) and add up to \( 2 \).
- The numbers \( 5 \) and \( -3 \) satisfy this condition because \( 5 \times (-3) = -15 \) and \( 5 + (-3) = 2 \).
3. **Write the factorization**:
\[ (y + 5)(y - 3) = 0 \]
4. **Set each factor to zero**:
- \( y + 5 = 0 \) gives \( y = -5 \)
- \( y - 3 = 0 \) gives \( y = 3 \)

### Step 3: List the solutions

From the above steps, we have:
- For the first equation: \( x = -4 \) and \( x = 3 \)
- For the second equation: \( y = -5 \) and \( y = 3 \)

### Step 4: Analyze the values of \( x \) and \( y \)

Now we compare the values of \( x \) and \( y \):
- \( x = -4 \) and \( y = -5 \): Here, \( -4 > -5 \)
- \( x = 3 \) and \( y = -5 \): Here, \( 3 > -5 \)
- \( x = 3 \) and \( y = 3 \): Here, \( x = y \)

### Conclusion

From the analysis, we have:
- \( -4 > -5 \)
- \( 3 > -5 \)
- \( 3 = 3 \)

Thus, we cannot establish a consistent relationship between all values of \( x \) and \( y \) since we have greater than, less than, and equal relationships. Therefore, the final answer is:

**No relation between \( x \) and \( y \)**.