Solve the given quadratic equations and mark the correct option based on your answer
I. ` x^(2) + 7 x + 12 = 0`
II. ` y^(2) + 9 y + 20 = 0`

Options

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

Answer

To solve the given quadratic equations, we will follow these steps:

### Step 1: Solve the first quadratic equation \( x^2 + 7x + 12 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 7 \), and \( c = 12 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( c = 12 \) and add up to \( b = 7 \). The numbers are \( 3 \) and \( 4 \).
3. **Rewrite the equation**:
\[ x^2 + 3x + 4x + 12 = 0 \]
4. **Group the terms**:
\[ (x^2 + 3x) + (4x + 12) = 0 \]
5. **Factor by grouping**:
\[ x(x + 3) + 4(x + 3) = 0 \]
\[ (x + 3)(x + 4) = 0 \]
6. **Set each factor to zero**:
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

### Step 2: Solve the second quadratic equation \( y^2 + 9y + 20 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 9 \), and \( c = 20 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( c = 20 \) and add up to \( b = 9 \). The numbers are \( 4 \) and \( 5 \).
3. **Rewrite the equation**:
\[ y^2 + 4y + 5y + 20 = 0 \]
4. **Group the terms**:
\[ (y^2 + 4y) + (5y + 20) = 0 \]
5. **Factor by grouping**:
\[ y(y + 4) + 5(y + 4) = 0 \]
\[ (y + 4)(y + 5) = 0 \]
6. **Set each factor to zero**:
\[ y + 4 = 0 \quad \Rightarrow \quad y = -4 \]
\[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \]

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

- The solutions for \( x \) are \( -3 \) and \( -4 \).
- The solutions for \( y \) are \( -4 \) and \( -5 \).

Now we can compare:

1. For \( x = -4 \) and \( y = -4 \): \( x = y \)
2. For \( x = -3 \) and \( y = -4 \): \( x > y \)
3. For \( x = -4 \) and \( y = -5 \): \( x > y \)
4. For \( x = -3 \) and \( y = -5 \): \( x > y \)

### Conclusion

From the comparisons, we can conclude that \( x \) is greater than or equal to \( y \). Therefore, the correct option is:

**Option B: \( x \geq y \)**

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