Solve the given quadratic equations and mark the correct option based on your answer
I. ` x^(2) - 14 x + 45 = 0`
II. ` y^(2) - 18 y + 72 = 0`

Options

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

Answer

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

### Step 1: Solve the first quadratic equation \( x^2 - 14x + 45 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -14 \), and \( c = 45 \).
2. **Find the factors of \( c \) (45)**: We need two numbers that multiply to \( 45 \) and add up to \( -14 \). The factors are \( -9 \) and \( -5 \).
3. **Rewrite the equation**:
\[ x^2 - 9x - 5x + 45 = 0 \]
4. **Factor by grouping**:
\[ x(x - 9) - 5(x - 9) = 0 \]
\[ (x - 9)(x - 5) = 0 \]
5. **Set each factor to zero**:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

### Step 2: Solve the second quadratic equation \( y^2 - 18y + 72 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -18 \), and \( c = 72 \).
2. **Find the factors of \( c \) (72)**: We need two numbers that multiply to \( 72 \) and add up to \( -18 \). The factors are \( -12 \) and \( -6 \).
3. **Rewrite the equation**:
\[ y^2 - 12y - 6y + 72 = 0 \]
4. **Factor by grouping**:
\[ y(y - 12) - 6(y - 12) = 0 \]
\[ (y - 12)(y - 6) = 0 \]
5. **Set each factor to zero**:
\[ y - 12 = 0 \quad \Rightarrow \quad y = 12 \]
\[ y - 6 = 0 \quad \Rightarrow \quad y = 6 \]

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

- The values obtained are \( x = 9 \) or \( x = 5 \) and \( y = 12 \) or \( y = 6 \).
- Comparing the values:
- \( 9 > 6 \) and \( 9 < 12 \)
- \( 5 < 6 \) and \( 5 < 12 \)

### Conclusion

From the comparisons:
- \( 5 < 6 < 9 < 12 \)

Thus, we can conclude that there is no consistent relationship between \( x \) and \( y \) that holds true for all cases. Therefore, the correct option is **D** (no relation can be established).