In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct options
I.` x^(2) + 29 x + 208 = 0 `
II.` y^(2) + 35 y + 306 = 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 equations given in the question, we will follow these steps:

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 29x + 208 = 0 \]

We need to factor this quadratic equation. We look for two numbers that multiply to \( 208 \) and add up to \( 29 \).

The numbers \( 16 \) and \( 13 \) satisfy these conditions because:
- \( 16 \times 13 = 208 \)
- \( 16 + 13 = 29 \)

Now we can rewrite the equation as:
\[ x^2 + 16x + 13x + 208 = 0 \]

### Step 2: Factor the Equation
Next, we can group the terms:
\[ (x^2 + 16x) + (13x + 208) = 0 \]

Factoring out the common terms gives us:
\[ x(x + 16) + 13(x + 16) = 0 \]

Now we can factor out \( (x + 16) \):
\[ (x + 16)(x + 13) = 0 \]

### Step 3: Find the Values of x
Setting each factor to zero gives us:
1. \( x + 16 = 0 \) → \( x = -16 \)
2. \( x + 13 = 0 \) → \( x = -13 \)

So the solutions for \( x \) are:
\[ x = -16 \quad \text{and} \quad x = -13 \]

### Step 4: Solve Equation II
The second equation is:
\[ y^2 + 35y + 306 = 0 \]

Again, we need to factor this quadratic equation. We look for two numbers that multiply to \( 306 \) and add up to \( 35 \).

The numbers \( 18 \) and \( 17 \) satisfy these conditions because:
- \( 18 \times 17 = 306 \)
- \( 18 + 17 = 35 \)

Now we can rewrite the equation as:
\[ y^2 + 18y + 17y + 306 = 0 \]

### Step 5: Factor the Equation
Next, we can group the terms:
\[ (y^2 + 18y) + (17y + 306) = 0 \]

Factoring out the common terms gives us:
\[ y(y + 18) + 17(y + 18) = 0 \]

Now we can factor out \( (y + 18) \):
\[ (y + 18)(y + 17) = 0 \]

### Step 6: Find the Values of y
Setting each factor to zero gives us:
1. \( y + 18 = 0 \) → \( y = -18 \)
2. \( y + 17 = 0 \) → \( y = -17 \)

So the solutions for \( y \) are:
\[ y = -18 \quad \text{and} \quad y = -17 \]

### Step 7: Compare Values of x and y
Now we have the values:
- For \( x \): \( -16 \) and \( -13 \)
- For \( y \): \( -18 \) and \( -17 \)

We compare each value of \( x \) with each value of \( y \):
1. Comparing \( x = -16 \) with \( y = -18 \):
- \( -16 > -18 \)
2. Comparing \( x = -16 \) with \( y = -17 \):
- \( -16 > -17 \)
3. Comparing \( x = -13 \) with \( y = -18 \):
- \( -13 > -18 \)
4. Comparing \( x = -13 \) with \( y = -17 \):
- \( -13 > -17 \)

In all comparisons, \( x \) is greater than \( y \).

### Conclusion
Thus, the relationship between \( x \) and \( y \) is:
\[ x > y \]