I.` x^(2) - 5x - 14 = 0" "II. Y^(2) + 7y + 10 = 0`

Options

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

Explanation

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

### Step 1: Solve the first equation \( x^2 - 5x - 14 = 0 \)

1. **Identify the coefficients**: The equation is in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -5 \), and \( c = -14 \).
2. **Find the product and sum**: We need two numbers that multiply to \( ac = 1 \times (-14) = -14 \) and add up to \( b = -5 \). The numbers are \( -7 \) and \( 2 \).
3. **Factor the equation**: Rewrite the equation as:
\[ x^2 - 7x + 2x - 14 = 0 \]
Grouping gives:
\[ x(x - 7) + 2(x - 7) = 0 \]
Factoring out \( (x - 7) \):
\[ (x - 7)(x + 2) = 0 \]
4. **Find the roots**: Set each factor to zero:
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Step 2: Solve the second equation \( y^2 + 7y + 10 = 0 \)

1. **Identify the coefficients**: The equation is in the form \( ay^2 + by + c = 0 \), where \( a = 1 \), \( b = 7 \), and \( c = 10 \).
2. **Find the product and sum**: We need two numbers that multiply to \( ac = 1 \times 10 = 10 \) and add up to \( b = 7 \). The numbers are \( 5 \) and \( 2 \).
3. **Factor the equation**: Rewrite the equation as:
\[ y^2 + 5y + 2y + 10 = 0 \]
Grouping gives:
\[ y(y + 5) + 2(y + 5) = 0 \]
Factoring out \( (y + 5) \):
\[ (y + 5)(y + 2) = 0 \]
4. **Find the roots**: Set each factor to zero:
\[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]

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

- The values of \( x \) are \( 7 \) and \( -2 \).
- The values of \( y \) are \( -5 \) and \( -2 \).

### Step 4: Analyze the relationships

- The value \( x = 7 \) is greater than both values of \( y \) (i.e., \( -5 \) and \( -2 \)).
- The value \( x = -2 \) is equal to \( y = -2 \).
- Therefore, we can conclude that \( x \) is greater than or equal to \( y \).

### Final Conclusion

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

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