In the following question two equations are given in variables x and y. you have to solve these equations and determine the relation between x and y.
I. `4x^(2) + 13x + 9 = 0`
II. `4y^(2) + 20y + 25 = 0`

Options

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

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation for \( x \)
The first equation is:
\[ 4x^2 + 13x + 9 = 0 \]
We will use the factorization method. We need to find two numbers that multiply to \( 4 \times 9 = 36 \) and add up to \( 13 \).

The numbers are \( 4 \) and \( 9 \). We can rewrite the equation as:
\[ 4x^2 + 4x + 9x + 9 = 0 \]
Now, we can group the terms:
\[ (4x^2 + 4x) + (9x + 9) = 0 \]
Factoring out the common terms:
\[ 4x(x + 1) + 9(x + 1) = 0 \]
Now, factor out \( (x + 1) \):
\[ (x + 1)(4x + 9) = 0 \]
Setting each factor to zero gives us:
1. \( x + 1 = 0 \) → \( x = -1 \)
2. \( 4x + 9 = 0 \) → \( 4x = -9 \) → \( x = -\frac{9}{4} = -2.25 \)

Thus, the solutions for \( x \) are:
\[ x_1 = -1 \quad \text{and} \quad x_2 = -2.25 \]

### Step 2: Solve the second equation for \( y \)
The second equation is:
\[ 4y^2 + 20y + 25 = 0 \]
Again, we will use the factorization method. We need to find two numbers that multiply to \( 4 \times 25 = 100 \) and add up to \( 20 \).

The numbers are \( 10 \) and \( 10 \). We can rewrite the equation as:
\[ 4y^2 + 10y + 10y + 25 = 0 \]
Now, we can group the terms:
\[ (4y^2 + 10y) + (10y + 25) = 0 \]
Factoring out the common terms:
\[ 2y(2y + 5) + 5(2y + 5) = 0 \]
Now, factor out \( (2y + 5) \):
\[ (2y + 5)(2y + 5) = 0 \]
Setting the factor to zero gives us:
\[ 2y + 5 = 0 \quad \Rightarrow \quad 2y = -5 \quad \Rightarrow \quad y = -\frac{5}{2} = -2.5 \]

Thus, the solution for \( y \) is:
\[ y = -2.5 \]

### Step 3: Determine the relationship between \( x \) and \( y \)
We have the values:
- \( x_1 = -1 \)
- \( x_2 = -2.25 \)
- \( y = -2.5 \)

Now, we compare:
1. \( -1 > -2.5 \)
2. \( -2.25 > -2.5 \)

In both cases, \( x \) is greater than \( y \).

### Conclusion
The relationship between \( x \) and \( y \) is:
\[ x > y \]