In the following questions two equations numbered I and II are given. You have to solve both the equations and Give answer
I. `x^(2)-7x+12=0` II. `y^(2)-12y+32=0`

Options

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

Answer

To solve the equations given in the question, we will follow these steps:

### Step 1: Solve the first equation \( x^2 - 7x + 12 = 0 \)

1. **Identify the equation**: We have the quadratic equation \( x^2 - 7x + 12 = 0 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( 12 \) (the constant term) and add up to \( -7 \) (the coefficient of \( x \)).
- The numbers that satisfy these conditions are \( -3 \) and \( -4 \).
3. **Rewrite the equation**: We can factor the equation as:
\[ (x - 3)(x - 4) = 0 \]
4. **Set each factor to zero**:
- \( x - 3 = 0 \) → \( x = 3 \)
- \( x - 4 = 0 \) → \( x = 4 \)

### Step 2: Solve the second equation \( y^2 - 12y + 32 = 0 \)

1. **Identify the equation**: We have the quadratic equation \( y^2 - 12y + 32 = 0 \).
2. **Factor the equation**: We need to find two numbers that multiply to \( 32 \) (the constant term) and add up to \( -12 \) (the coefficient of \( y \)).
- The numbers that satisfy these conditions are \( -4 \) and \( -8 \).
3. **Rewrite the equation**: We can factor the equation as:
\[ (y - 4)(y - 8) = 0 \]
4. **Set each factor to zero**:
- \( y - 4 = 0 \) → \( y = 4 \)
- \( y - 8 = 0 \) → \( y = 8 \)

### Step 3: Summarize the solutions

- The solutions for \( x \) are \( x = 3 \) and \( x = 4 \).
- The solutions for \( y \) are \( y = 4 \) and \( y = 8 \).

### Step 4: Compare the values of \( x \) and \( y \)

From the solutions:
- When \( x = 4 \), \( y = 4 \) (they are equal).
- When \( x = 3 \), \( y = 8 \) (here \( y > x \)).

### Conclusion

The relationship between \( x \) and \( y \) can be summarized as:
- \( y \) is either equal to \( x \) or greater than \( x \).
- Therefore, we can conclude that \( y \geq x \).