` 2x^(2) - 15x + 28 = 0 `
` 4y^(2) - 23y+30 = 0`

Options

A. ` x gt y`
B. ` x lt y`
C. ` x ge y`
D. x = y or relation cannot be established

Explanation

To solve the equations \(2x^2 - 15x + 28 = 0\) and \(4y^2 - 23y + 30 = 0\), we will follow these steps:

### Step 1: Solve the first equation \(2x^2 - 15x + 28 = 0\)

We can factor the quadratic equation. We will look for two numbers that multiply to \(2 \times 28 = 56\) and add up to \(-15\).

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

Now, we can rewrite the equation:

\[ 2x^2 - 8x - 7x + 28 = 0 \]

Next, we can group the terms:

\[ (2x^2 - 8x) + (-7x + 28) = 0 \]

Factoring out the common terms gives us:

\[ 2x(x - 4) - 7(x - 4) = 0 \]

Now, we can factor out \((x - 4)\):

\[ (2x - 7)(x - 4) = 0 \]

Setting each factor to zero gives us the solutions:

1. \(2x - 7 = 0 \implies x = \frac{7}{2} = 3.5\)
2. \(x - 4 = 0 \implies x = 4\)

### Step 2: Solve the second equation \(4y^2 - 23y + 30 = 0\)

Similarly, we will factor this quadratic equation. We need two numbers that multiply to \(4 \times 30 = 120\) and add up to \(-23\).

The numbers that satisfy this are \(-15\) and \(-8\).

Now, we can rewrite the equation:

\[ 4y^2 - 15y - 8y + 30 = 0 \]

Next, we can group the terms:

\[ (4y^2 - 15y) + (-8y + 30) = 0 \]

Factoring out the common terms gives us:

\[ y(4y - 15) - 2(4y - 15) = 0 \]

Now, we can factor out \((4y - 15)\):

\[ (4y - 15)(y - 2) = 0 \]

Setting each factor to zero gives us the solutions:

1. \(4y - 15 = 0 \implies y = \frac{15}{4} = 3.75\)
2. \(y - 2 = 0 \implies y = 2\)

### Step 3: Summary of Solutions

The solutions for \(x\) are:
- \(x = 3.5\)
- \(x = 4\)

The solutions for \(y\) are:
- \(y = 3.75\)
- \(y = 2\)

### Step 4: Determine the relationship between \(x\) and \(y\)

From the solutions, we can compare the values:
- \(3.5 < 3.75\)
- \(4 > 3.75\)
- \(4 > 2\)
- \(3.5 > 2\)

### Conclusion

The relationship between \(x\) and \(y\) can be summarized as follows:
- \(x\) can be greater than or less than \(y\) depending on the specific values chosen.