Solve the given quadratic equations and mark the correct options based on your answer
I. ` x^(2) + 7 x + 10 = 0 `
II. ` 2 y ^(2) + 5y + 2 = 0 `

Options

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

Answer

To solve the given quadratic equations and find the relationship between the solutions, we will follow these steps:

### Step 1: Solve the first equation \( x^2 + 7x + 10 = 0 \)

1. **Factor the quadratic equation**:
\[ x^2 + 7x + 10 = (x + 5)(x + 2) = 0 \]

2. **Set each factor to zero**:
\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

**Solutions for the first equation**: \( x = -5 \) and \( x = -2 \).

### Step 2: Solve the second equation \( 2y^2 + 5y + 2 = 0 \)

1. **Factor the quadratic equation**:
\[ 2y^2 + 5y + 2 = (2y + 1)(y + 2) = 0 \]

2. **Set each factor to zero**:
\[ 2y + 1 = 0 \quad \Rightarrow \quad y = -\frac{1}{2} \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]

**Solutions for the second equation**: \( y = -2 \) and \( y = -\frac{1}{2} \).

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

- The solutions for \( x \) are \( -5 \) and \( -2 \).
- The solutions for \( y \) are \( -2 \) and \( -\frac{1}{2} \).

### Step 4: Establish relationships between \( x \) and \( y \)

1. **Comparing \( x = -5 \)**:
- \( -5 < -2 \) (True)
- \( -5 < -\frac{1}{2} \) (True)

2. **Comparing \( x = -2 \)**:
- \( -2 = -2 \) (True)
- \( -2 > -\frac{1}{2} \) (True)

### Conclusion

From the comparisons, we can summarize:
- For \( x = -5 \), \( x < y \) for both values of \( y \).
- For \( x = -2 \), \( x = y \) when \( y = -2 \).

Thus, the overall relationship is:
\[ x \leq y \]

### Final Answer
The correct option based on the solutions is:
**Option D: \( x \leq y \)**.

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