In each question, two equations numbered I and II have been given. You have to solve both the equations and mark the appropriate option
I`x^2+11x+18=0`
II`y^2+16y+48=0`

Options

A. `xgty`
B. `xley`
C. no relation can be established between x and y
D. `xgey`

Answer

To solve the given equations step by step, we will first address each quadratic equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 11x + 18 = 0 \]

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \( 18 \) (the constant term) and add up to \( 11 \) (the coefficient of \( x \)).

The numbers that satisfy this are \( 9 \) and \( 2 \). Thus, we can factor the equation as:
\[ (x + 9)(x + 2) = 0 \]

Setting each factor to zero gives us:
1. \( x + 9 = 0 \) → \( x = -9 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

So the solutions for Equation I are:
\[ x = -9 \text{ and } x = -2 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 + 16y + 48 = 0 \]

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 48 \) and add up to \( 16 \).

The numbers that satisfy this are \( 12 \) and \( 4 \). Thus, we can factor the equation as:
\[ (y + 12)(y + 4) = 0 \]

Setting each factor to zero gives us:
1. \( y + 12 = 0 \) → \( y = -12 \)
2. \( y + 4 = 0 \) → \( y = -4 \)

So the solutions for Equation II are:
\[ y = -12 \text{ and } y = -4 \]

### Step 3: Analyze the Relationship Between x and y
Now we have the solutions:
- From Equation I: \( x = -9 \) and \( x = -2 \)
- From Equation II: \( y = -12 \) and \( y = -4 \)

Next, we will compare the values of \( x \) and \( y \):
- \( x = -9 \) is less than \( y = -4 \)
- \( x = -2 \) is greater than \( y = -12 \)

Since there is no consistent relationship between the values of \( x \) and \( y \) (one is greater in one case and lesser in another), we conclude that:
**There is no relationship between \( x \) and \( y \).**

### Final Answer
The correct answer is:
**Option 3: No relation can be established between \( x \) and \( y \).**