(i) ` x^(2) - 5x-14 = 0`
(ii) ` 2y^(2) +11y + 14 = 0`

Options

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

Explanation

Let's solve the equations step by step.

### Part (i): Solve the equation \( x^2 - 5x - 14 = 0 \)

1. **Identify the equation**: We have the quadratic equation \( x^2 - 5x - 14 = 0 \).

2. **Factor the quadratic**: We need to find two numbers that multiply to \(-14\) (the constant term) and add up to \(-5\) (the coefficient of \(x\)). The numbers \(-7\) and \(2\) satisfy this condition because:
- \(-7 \times 2 = -14\)
- \(-7 + 2 = -5\)

3. **Rewrite the equation**: We can rewrite the equation as:
\[ x^2 - 7x + 2x - 14 = 0 \]

4. **Group the terms**: Group the first two and the last two terms:
\[ (x^2 - 7x) + (2x - 14) = 0 \]

5. **Factor by grouping**: Factor out the common terms:
\[ x(x - 7) + 2(x - 7) = 0 \]
This can be factored further as:
\[ (x - 7)(x + 2) = 0 \]

6. **Set each factor to zero**: Now, set each factor equal to zero:
\[ x - 7 = 0 \quad \text{or} \quad x + 2 = 0 \]

7. **Solve for \(x\)**:
\[ x = 7 \quad \text{or} \quad x = -2 \]

### Part (ii): Solve the equation \( 2y^2 + 11y + 14 = 0 \)

1. **Identify the equation**: We have the quadratic equation \( 2y^2 + 11y + 14 = 0 \).

2. **Multiply the leading coefficient and the constant**: Multiply \(2\) (the coefficient of \(y^2\)) by \(14\) (the constant term) to get \(28\).

3. **Find factors of \(28\)**: We need to find two numbers that multiply to \(28\) and add up to \(11\). The numbers \(7\) and \(4\) satisfy this condition because:
- \(7 \times 4 = 28\)
- \(7 + 4 = 11\)

4. **Rewrite the equation**: We can rewrite the equation as:
\[ 2y^2 + 7y + 4y + 14 = 0 \]

5. **Group the terms**: Group the first two and the last two terms:
\[ (2y^2 + 7y) + (4y + 14) = 0 \]

6. **Factor by grouping**: Factor out the common terms:
\[ y(2y + 7) + 2(2y + 7) = 0 \]
This can be factored further as:
\[ (2y + 7)(y + 2) = 0 \]

7. **Set each factor to zero**: Now, set each factor equal to zero:
\[ 2y + 7 = 0 \quad \text{or} \quad y + 2 = 0 \]

8. **Solve for \(y\)**:
\[ 2y = -7 \quad \Rightarrow \quad y = -\frac{7}{2} \]
\[ y = -2 \]

### Summary of Solutions
- The solutions for \(x\) are \(x = 7\) and \(x = -2\).
- The solutions for \(y\) are \(y = -2\) and \(y = -\frac{7}{2}\).