(i)` x^(2) -10x+21 = 0`
(ii) ` y^(2) -11y+18 = 0`

Options

A. ` x lt y`
B. x =y or no relation can be established between x and y.
C. ` x ge y`
D. ` x gt y `

Explanation

To solve the equations step by step, we will start with each equation separately.

### Step 1: Solve the first equation \( x^2 - 10x + 21 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -10 \), and \( c = 21 \).
2. **Factor the quadratic**: We need two numbers that multiply to \( c = 21 \) and add up to \( b = -10 \). The numbers are \( -3 \) and \( -7 \).
3. **Rewrite the equation**:
\[ x^2 - 3x - 7x + 21 = 0 \]
This can be factored as:
\[ (x - 3)(x - 7) = 0 \]
4. **Set each factor to zero**:
\[ x - 3 = 0 \quad \text{or} \quad x - 7 = 0 \]
5. **Solve for \( x \)**:
\[ x = 3 \quad \text{or} \quad x = 7 \]

### Step 2: Solve the second equation \( y^2 - 11y + 18 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -11 \), and \( c = 18 \).
2. **Factor the quadratic**: We need two numbers that multiply to \( c = 18 \) and add up to \( b = -11 \). The numbers are \( -2 \) and \( -9 \).
3. **Rewrite the equation**:
\[ y^2 - 2y - 9y + 18 = 0 \]
This can be factored as:
\[ (y - 2)(y - 9) = 0 \]
4. **Set each factor to zero**:
\[ y - 2 = 0 \quad \text{or} \quad y - 9 = 0 \]
5. **Solve for \( y \)**:
\[ y = 2 \quad \text{or} \quad y = 9 \]

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

- The values of \( x \) are \( 3 \) and \( 7 \).
- The values of \( y \) are \( 2 \) and \( 9 \).

### Step 4: Establish relationships

1. Compare \( x \) and \( y \):
- \( 3 \) is greater than \( 2 \) but less than \( 9 \).
- \( 7 \) is greater than \( 2 \) but less than \( 9 \).

2. Conclusion:
- There is no direct relationship established between \( x \) and \( y \) in terms of one being equal to the other.

### Final Result:

- The values of \( x \) are \( 3 \) and \( 7 \).
- The values of \( y \) are \( 2 \) and \( 9 \).
- There is no relation between \( x \) and \( y \).

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