In this question two equations numbered I and II are given. You have to solve both the equations and mark the appropriate option. Give answer.
I `x^2+8x+15=0`
II `y^2+13y+40=0`

Options

A. `xgty`
B. `xlty`
C. `xley`
D. `xgey`

Answer

To solve the given equations step by step, we will start with the first equation and then move to the second one.

### Step 1: Solve the first equation \( x^2 + 8x + 15 = 0 \)

1. **Identify the equation**: The equation is \( x^2 + 8x + 15 = 0 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( 15 \) (the constant term) and add up to \( 8 \) (the coefficient of \( x \)). These numbers are \( 3 \) and \( 5 \).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ (x + 3)(x + 5) = 0 \]
4. **Set each factor to zero**:
- \( x + 3 = 0 \) → \( x = -3 \)
- \( x + 5 = 0 \) → \( x = -5 \)

### Step 2: Solve the second equation \( y^2 + 13y + 40 = 0 \)

1. **Identify the equation**: The equation is \( y^2 + 13y + 40 = 0 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( 40 \) and add up to \( 13 \). These numbers are \( 5 \) and \( 8 \).
3. **Rewrite the equation**: We can rewrite the equation as:
\[ (y + 5)(y + 8) = 0 \]
4. **Set each factor to zero**:
- \( y + 5 = 0 \) → \( y = -5 \)
- \( y + 8 = 0 \) → \( y = -8 \)

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

- From the first equation, we found \( x = -3 \) and \( x = -5 \).
- From the second equation, we found \( y = -5 \) and \( y = -8 \).

### Step 4: Determine the relationship between \( x \) and \( y \)

- The values of \( x \) are \( -3 \) and \( -5 \).
- The values of \( y \) are \( -5 \) and \( -8 \).
- We compare the maximum values of \( x \) and \( y \):
- The maximum value of \( x \) is \( -3 \).
- The maximum value of \( y \) is \( -5 \).

Since \( -3 > -5 \), we conclude that:
\[ x > y \]

### Final Answer

The correct relationship is \( x \geq y \).

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