In each of the following questions two equations are given. Solve these equations and give answer
I. ` x^(2) - 8 x + 15 = 0 `
II. ` 2 y ^(2) - 5y - 3 = 0 `

Options

A. A)If ` x ge y ` , i.e. x is greater than
or equal to y
B. B)If ` x gt y `, i.e. x is greater than y
C. C)If ` x le y`, i.e. x is less than
or equal to y
D. D) If ` x lt y ` , i.e. x is less than y

Answer

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

### Step 1: Solve the first equation \( x^2 - 8x + 15 = 0 \)

1. **Identify the equation**: We have \( x^2 - 8x + 15 = 0 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \( 15 \) (the constant term) and add up to \( -8 \) (the coefficient of \( x \)).
- The numbers are \( -5 \) and \( -3 \).
3. **Rewrite the equation**:
\[ (x - 5)(x - 3) = 0 \]
4. **Set each factor to zero**:
- \( x - 5 = 0 \) gives \( x = 5 \)
- \( x - 3 = 0 \) gives \( x = 3 \)

### Step 2: Solve the second equation \( 2y^2 - 5y - 3 = 0 \)

1. **Identify the equation**: We have \( 2y^2 - 5y - 3 = 0 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \( 2 \times -3 = -6 \) and add up to \( -5 \).
- The numbers are \( -6 \) and \( 1 \).
3. **Rewrite the equation**: Break the middle term:
\[ 2y^2 - 6y + y - 3 = 0 \]
4. **Group the terms**:
\[ (2y^2 - 6y) + (y - 3) = 0 \]
5. **Factor by grouping**:
- From the first group, factor out \( 2y \):
\[ 2y(y - 3) + 1(y - 3) = 0 \]
- Now factor out \( (y - 3) \):
\[ (y - 3)(2y + 1) = 0 \]
6. **Set each factor to zero**:
- \( y - 3 = 0 \) gives \( y = 3 \)
- \( 2y + 1 = 0 \) gives \( y = -\frac{1}{2} \)

### Summary of Solutions

- From the first equation, we found \( x = 5 \) and \( x = 3 \).
- From the second equation, we found \( y = 3 \) and \( y = -\frac{1}{2} \).

### Final Relations

1. If \( x = 5 \) and \( y = 3 \), then \( x > y \).
2. If \( x = 3 \) and \( y = 3 \), then \( x = y \).

### Conclusion

The possible relations are:
- \( x > y \) when \( x = 5 \) and \( y = 3 \).
- \( x = y \) when both are equal to \( 3 \).

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