In each of the following questions, two equations (I) and (II) are given. Solve the equations and mark the correct option
I. ` x^(2) + 12 x + 35 = 0 `
II. ` y^(2) + 14 y + 48 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y `
C. if ` x lt y `
D. If x = y or no relation can be established between x and y

Answer

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

### Step 1: Solve the first equation \( x^2 + 12x + 35 = 0 \)

We need to factor the quadratic equation. We are looking for two numbers that multiply to \( 35 \) (the constant term) and add up to \( 12 \) (the coefficient of \( x \)).

The numbers that satisfy this condition are \( 7 \) and \( 5 \) because:
- \( 7 \times 5 = 35 \)
- \( 7 + 5 = 12 \)

So we can rewrite the equation as:
\[ x^2 + 7x + 5x + 35 = 0 \]

### Step 2: Factor the equation

Now, we can group the terms:
\[ (x^2 + 7x) + (5x + 35) = 0 \]
Factoring by grouping gives us:
\[ x(x + 7) + 5(x + 7) = 0 \]
Now, we can factor out the common term \( (x + 7) \):
\[ (x + 7)(x + 5) = 0 \]

### Step 3: Find the values of \( x \)

Setting each factor to zero gives us:
1. \( x + 7 = 0 \) → \( x = -7 \)
2. \( x + 5 = 0 \) → \( x = -5 \)

So, the solutions for \( x \) are:
\[ x = -7 \quad \text{and} \quad x = -5 \]

### Step 4: Solve the second equation \( y^2 + 14y + 48 = 0 \)

Similarly, we need to factor this quadratic equation. We are looking for two numbers that multiply to \( 48 \) and add up to \( 14 \).

The numbers that satisfy this condition are \( 8 \) and \( 6 \) because:
- \( 8 \times 6 = 48 \)
- \( 8 + 6 = 14 \)

So we can rewrite the equation as:
\[ y^2 + 8y + 6y + 48 = 0 \]

### Step 5: Factor the equation

Now, we can group the terms:
\[ (y^2 + 8y) + (6y + 48) = 0 \]
Factoring by grouping gives us:
\[ y(y + 8) + 6(y + 8) = 0 \]
Now, we can factor out the common term \( (y + 8) \):
\[ (y + 8)(y + 6) = 0 \]

### Step 6: Find the values of \( y \)

Setting each factor to zero gives us:
1. \( y + 8 = 0 \) → \( y = -8 \)
2. \( y + 6 = 0 \) → \( y = -6 \)

So, the solutions for \( y \) are:
\[ y = -8 \quad \text{and} \quad y = -6 \]

### Step 7: Compare the values of \( x \) and \( y \)

Now we have the values:
- For \( x \): \( -7 \) and \( -5 \)
- For \( y \): \( -8 \) and \( -6 \)

We can compare:
- \( -7 \) is greater than \( -8 \) and less than \( -6 \)
- \( -5 \) is greater than both \( -8 \) and \( -6 \)

### Conclusion

From the comparisons, we can see that:
- \( -7 > -8 \) and \( -7 < -6 \)
- \( -5 > -8 \) and \( -5 > -6 \)

Thus, we cannot establish a consistent relationship between \( x \) and \( y \).

### Final Answer

The correct option is:
**D: No relation can be established between \( x \) and \( y \)**.