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. `2x^(2) + 15x + 28 =0`
II. `2y^(2) + 19 y + 45 = 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:
\[ 2x^2 + 15x + 28 = 0 \]

To solve this quadratic equation, we can use the factorization method. We need to find two numbers that multiply to \( 2 \times 28 = 56 \) and add up to \( 15 \).

The numbers that satisfy this condition are \( 8 \) and \( 7 \).

We can rewrite the equation as:
\[ 2x^2 + 8x + 7x + 28 = 0 \]

Now, we can group the terms:
\[ (2x^2 + 8x) + (7x + 28) = 0 \]

Factoring out the common terms:
\[ 2x(x + 4) + 7(x + 4) = 0 \]

Now, factor out \( (x + 4) \):
\[ (2x + 7)(x + 4) = 0 \]

Setting each factor to zero gives us:
1. \( 2x + 7 = 0 \) → \( x = -\frac{7}{2} = -3.5 \)
2. \( x + 4 = 0 \) → \( x = -4 \)

So, the values of \( x \) are \( -3.5 \) and \( -4 \).

### Step 2: Solve the second equation for \( y \)
The second equation is:
\[ 2y^2 + 19y + 45 = 0 \]

Again, we will use the factorization method. We need to find two numbers that multiply to \( 2 \times 45 = 90 \) and add up to \( 19 \).

The numbers that satisfy this condition are \( 9 \) and \( 10 \).

We can rewrite the equation as:
\[ 2y^2 + 9y + 10y + 45 = 0 \]

Now, we can group the terms:
\[ (2y^2 + 9y) + (10y + 45) = 0 \]

Factoring out the common terms:
\[ y(2y + 9) + 5(2y + 9) = 0 \]

Now, factor out \( (2y + 9) \):
\[ (2y + 9)(y + 5) = 0 \]

Setting each factor to zero gives us:
1. \( 2y + 9 = 0 \) → \( y = -\frac{9}{2} = -4.5 \)
2. \( y + 5 = 0 \) → \( y = -5 \)

So, the values of \( y \) are \( -4.5 \) and \( -5 \).

### Step 3: Determine the relationship between \( x \) and \( y \)
Now we have the following values:
- For \( x \): \( -3.5 \) and \( -4 \)
- For \( y \): \( -4.5 \) and \( -5 \)

Now, we can compare the values:
- \( -3.5 > -4.5 \)
- \( -3.5 > -5 \)
- \( -4 > -4.5 \)
- \( -4 > -5 \)

From the comparisons, we can conclude that:
- In both cases, \( x \) is greater than \( y \).

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