Solve the given quadratic equations and mark the correct option based on your answer -
I. ` x^(2) - 11 x + 30 = 0`
II. ` y^(2) - 15 y + 56 = 0`

Options

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

Answer

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

### Step 1: Solve the first equation \( x^2 - 11x + 30 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -11 \), and \( c = 30 \).
2. **Factor the quadratic**: We need two numbers that multiply to \( c \) (30) and add up to \( b \) (-11). The numbers are -5 and -6.
3. **Rewrite the equation**:
\[ x^2 - 5x - 6x + 30 = 0 \]
This can be factored as:
\[ (x - 5)(x - 6) = 0 \]
4. **Find the roots**: Set each factor to zero:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]

### Step 2: Solve the second equation \( y^2 - 15y + 56 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -15 \), and \( c = 56 \).
2. **Factor the quadratic**: We need two numbers that multiply to \( c \) (56) and add up to \( b \) (-15). The numbers are -7 and -8.
3. **Rewrite the equation**:
\[ y^2 - 7y - 8y + 56 = 0 \]
This can be factored as:
\[ (y - 7)(y - 8) = 0 \]
4. **Find the roots**: Set each factor to zero:
\[ y - 7 = 0 \quad \Rightarrow \quad y = 7 \]
\[ y - 8 = 0 \quad \Rightarrow \quad y = 8 \]

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

- The values of \( x \) are 5 and 6.
- The values of \( y \) are 7 and 8.

Now, we will compare the values of \( x \) and \( y \):

1. For \( x = 5 \):
- \( 5 < 7 \) and \( 5 < 8 \)
2. For \( x = 6 \):
- \( 6 < 7 \) and \( 6 < 8 \)

From both comparisons, we can conclude that \( x < y \).

### Final Answer

Thus, the relation we have is:
\[ x < y \]