In each of these questions, two equations I. and II are given. You have to solve both the equations and give answer
I. ` x^(2) - 4 x - 5 = 0 `
II. ` 7 y^(2) - 25 y - 12 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y`
C. If ` x lt y `
D. if x = y or relationship between x and y cannot be established

Answer

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

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 4x - 5 = 0 \]

This is a quadratic equation. We can factor it to find the values of \( x \).

1. **Identify factors**: We need two numbers that multiply to \(-5\) (the constant term) and add up to \(-4\) (the coefficient of \( x \)).
- The factors of \(-5\) that satisfy this condition are \(-5\) and \(1\).

2. **Rewrite the equation**:
\[ x^2 - 5x + 1x - 5 = 0 \]
This can be grouped as:
\[ (x^2 - 5x) + (1x - 5) = 0 \]

3. **Factor by grouping**:
\[ x(x - 5) + 1(x - 5) = 0 \]
\[ (x - 5)(x + 1) = 0 \]

4. **Set each factor to zero**:
\[ x - 5 = 0 \quad \text{or} \quad x + 1 = 0 \]

5. **Solve for \( x \)**:
\[ x = 5 \quad \text{or} \quad x = -1 \]

### Step 2: Solve Equation II
The second equation is:
\[ 7y^2 - 25y - 12 = 0 \]

This is also a quadratic equation. We will use the factorization method.

1. **Identify factors**: We need two numbers that multiply to \(7 \times -12 = -84\) and add up to \(-25\).
- The factors of \(-84\) that satisfy this condition are \(-28\) and \(3\).

2. **Rewrite the equation**:
\[ 7y^2 - 28y + 3y - 12 = 0 \]
This can be grouped as:
\[ (7y^2 - 28y) + (3y - 12) = 0 \]

3. **Factor by grouping**:
\[ 7y(y - 4) + 3(y - 4) = 0 \]
\[ (y - 4)(7y + 3) = 0 \]

4. **Set each factor to zero**:
\[ y - 4 = 0 \quad \text{or} \quad 7y + 3 = 0 \]

5. **Solve for \( y \)**:
\[ y = 4 \quad \text{or} \quad y = -\frac{3}{7} \]

### Step 3: Summary of Solutions
From the two equations, we have:
- From Equation I: \( x = 5 \) or \( x = -1 \)
- From Equation II: \( y = 4 \) or \( y = -\frac{3}{7} \)

### Step 4: Compare Values
Now, we will compare the values of \( x \) and \( y \):

1. **Comparing \( x = 5 \) and \( y = 4 \)**:
\[ 5 > 4 \quad (\text{So } x > y) \]

2. **Comparing \( x = 5 \) and \( y = -\frac{3}{7} \)**:
\[ 5 > -\frac{3}{7} \quad (\text{So } x > y) \]

3. **Comparing \( x = -1 \) and \( y = 4 \)**:
\[ -1 < 4 \quad (\text{So } x < y) \]

4. **Comparing \( x = -1 \) and \( y = -\frac{3}{7} \)**:
\[ -1 < -\frac{3}{7} \quad (\text{So } x < y) \]

### Conclusion
The relationships established are:
- For \( x = 5 \), \( x > y \) in both cases.
- For \( x = -1 \), \( x < y \) in both cases.

Thus, there is no consistent relationship between \( x \) and \( y \) across all values. Therefore, we conclude that the relationship between \( x \) and \( y \) cannot be established.