Given below are two equations in each question, which you have to solve and give answer
I. ` x^(2) - 3 x = `4
II. ` y^(2) + 6y + 8 = 0 `

Options

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

Answer

To solve the given equations step by step, we will tackle each equation separately.

### Step 1: Solve the first equation \( x^2 - 3x = 4 \)

1. **Rearrange the equation**:
\[ x^2 - 3x - 4 = 0 \]
(We move 4 to the left side of the equation.)

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(-4\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(-4\) and \(1\) work:
\[ (x - 4)(x + 1) = 0 \]

3. **Set each factor to zero**:
\[ x - 4 = 0 \quad \text{or} \quad x + 1 = 0 \]
This gives us:
\[ x = 4 \quad \text{or} \quad x = -1 \]

### Step 2: Solve the second equation \( y^2 + 6y + 8 = 0 \)

1. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(8\) and add to \(6\). The numbers \(4\) and \(2\) work:
\[ (y + 4)(y + 2) = 0 \]

2. **Set each factor to zero**:
\[ y + 4 = 0 \quad \text{or} \quad y + 2 = 0 \]
This gives us:
\[ y = -4 \quad \text{or} \quad y = -2 \]

### Summary of Solutions
- The values of \(x\) are \(4\) and \(-1\).
- The values of \(y\) are \(-4\) and \(-2\).

### Step 3: Compare the values of \(x\) and \(y\)

1. Compare \(x = 4\) with \(y = -4\):
\[ 4 > -4 \]

2. Compare \(x = 4\) with \(y = -2\):
\[ 4 > -2 \]

3. Compare \(x = -1\) with \(y = -4\):
\[ -1 > -4 \]

4. Compare \(x = -1\) with \(y = -2\):
\[ -1 > -2 \]

### Conclusion
In all comparisons, \(x\) is greater than \(y\). Therefore, the final answer is:
\[ \text{X is greater than Y} \]

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