Solve the equations given below and give answer
1) if `x lt y` 2) if `x gt y`
3) if `c ge y` 4) if `x le y`
5) if x=y or no relation can be established
I. `2x^2+9x+10=0` II. `4y^2+28y+45=0`

Options

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

Answer

To solve the 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:
\[ 2x^2 + 9x + 10 = 0 \]

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \(2 \times 10 = 20\) and add up to \(9\). The numbers \(4\) and \(5\) work since \(4 + 5 = 9\) and \(4 \times 5 = 20\).

Now we can rewrite the equation:
\[ 2x^2 + 4x + 5x + 10 = 0 \]
Grouping the terms:
\[ (2x^2 + 4x) + (5x + 10) = 0 \]
Factoring out the common terms:
\[ 2x(x + 2) + 5(x + 2) = 0 \]
Now factor out \((x + 2)\):
\[ (2x + 5)(x + 2) = 0 \]

Setting each factor to zero gives us:
1. \(2x + 5 = 0 \Rightarrow x = -\frac{5}{2} = -2.5\)
2. \(x + 2 = 0 \Rightarrow x = -2\)

So, the values of \(x\) are:
\[ x_1 = -2.5 \quad \text{and} \quad x_2 = -2 \]

### Step 2: Solve the second equation for \(y\)
The second equation is:
\[ 4y^2 + 28y + 45 = 0 \]

Again, we will factor this quadratic equation. We need two numbers that multiply to \(4 \times 45 = 180\) and add to \(28\). The numbers \(18\) and \(10\) work since \(18 + 10 = 28\) and \(18 \times 10 = 180\).

Rewriting the equation:
\[ 4y^2 + 18y + 10y + 45 = 0 \]
Grouping the terms:
\[ (4y^2 + 18y) + (10y + 45) = 0 \]
Factoring out the common terms:
\[ 2y(2y + 9) + 5(2y + 9) = 0 \]
Now factor out \((2y + 9)\):
\[ (2y + 9)(2y + 5) = 0 \]

Setting each factor to zero gives us:
1. \(2y + 9 = 0 \Rightarrow y = -\frac{9}{2} = -4.5\)
2. \(2y + 5 = 0 \Rightarrow y = -\frac{5}{2} = -2.5\)

So, the values of \(y\) are:
\[ y_1 = -4.5 \quad \text{and} \quad y_2 = -2.5 \]

### Step 3: Compare the values of \(x\) and \(y\)
Now we have the following values:
- For \(x\): \(-2.5\) and \(-2\)
- For \(y\): \(-4.5\) and \(-2.5\)

We will analyze the relationships:

1. **If \(x = -2\) and \(y = -4.5\)**:
- Here, \(-2 > -4.5\) (so \(x > y\))

2. **If \(x = -2.5\) and \(y = -2.5\)**:
- Here, \(x = y\)

3. **If \(x = -2.5\) and \(y = -4.5\)**:
- Here, \(-2.5 > -4.5\) (so \(x > y\))

4. **If \(x = -2\) and \(y = -2.5\)**:
- Here, \(-2 > -2.5\) (so \(x > y\))

### Conclusion
From the comparisons, we can conclude:
- In all cases except when \(x = y\), \(x\) is greater than \(y\).

Thus, the final relationship can be summarized as:
- \(x \geq y\)

### Final Answer
The answer is that \(x\) is greater than or equal to \(y\).

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