Solve the given quadratic equations and mark the correct option based on your answer -
I. ` 4 x^(2) - 24 x + 32 = 0`
II. ` y^(2) - 8y + 15 = 0`

Options

A. ` x gt y`
B. ` x lt y`
C. ` x ge y `
D. x = y or there is no relationship

Answer

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

### Step 1: Solve the first equation \( 4x^2 - 24x + 32 = 0 \)

1. **Divide the entire equation by 4** to simplify it:
\[ x^2 - 6x + 8 = 0 \]

**Hint:** Dividing by a common factor simplifies the equation.

2. **Factor the quadratic equation**:
We need two numbers that multiply to \(8\) (the constant term) and add up to \(-6\) (the coefficient of \(x\)). The numbers are \(-4\) and \(-2\):
\[ (x - 4)(x - 2) = 0 \]

**Hint:** Look for pairs of factors of the constant term that add up to the middle coefficient.

3. **Set each factor to zero**:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

**Hint:** Setting each factor equal to zero gives the possible solutions for \(x\).

### Step 2: Solve the second equation \( y^2 - 8y + 15 = 0 \)

1. **Factor the quadratic equation**:
We need two numbers that multiply to \(15\) and add up to \(-8\). The numbers are \(-5\) and \(-3\):
\[ (y - 5)(y - 3) = 0 \]

**Hint:** Similar to the first equation, find pairs of factors for the constant term.

2. **Set each factor to zero**:
\[ y - 5 = 0 \quad \Rightarrow \quad y = 5 \]
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]

**Hint:** Again, setting each factor equal to zero gives the possible solutions for \(y\).

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

We found:
- \(x = 4\) or \(x = 2\)
- \(y = 5\) or \(y = 3\)

Now we compare the values:
- For \(x = 4\):
- \(4 < 5\) (True)
- \(4 > 3\) (True)

- For \(x = 2\):
- \(2 < 5\) (True)
- \(2 < 3\) (True)

### Conclusion

From the comparisons:
- In the case of \(x = 4\), \(x\) is less than \(y\) when \(y = 5\) and greater when \(y = 3\).
- In the case of \(x = 2\), \(x\) is less than both values of \(y\).

Thus, we cannot establish a consistent relationship between \(x\) and \(y\) across all comparisons. Therefore, the conclusion is:

**No relation exists between \(x\) and \(y\).**