Solve the given quadratic equations and mark the correct option based on your answer -
I. ` x^(2) - 21 x + 108 = 0`
II. ` y^(2) - 17 y + 72 = 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 and determine the relationship between the values of \(x\) and \(y\), we will follow these steps:

### Step 1: Solve the first quadratic equation \(x^2 - 21x + 108 = 0\)

We will factor the quadratic equation. We need to find two numbers that multiply to \(108\) (the constant term) and add up to \(-21\) (the coefficient of \(x\)).

The factors of \(108\) that add up to \(21\) are \(12\) and \(9\).

Thus, we can write:
\[ x^2 - 21x + 108 = (x - 12)(x - 9) = 0 \]

### Step 2: Find the roots of the first equation

Setting each factor to zero gives us:
1. \(x - 12 = 0 \Rightarrow x = 12\)
2. \(x - 9 = 0 \Rightarrow x = 9\)

So, the solutions for \(x\) are:
\[ x = 12 \quad \text{or} \quad x = 9 \]

### Step 3: Solve the second quadratic equation \(y^2 - 17y + 72 = 0\)

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \(72\) and add up to \(-17\).

The factors of \(72\) that add up to \(17\) are \(8\) and \(9\).

Thus, we can write:
\[ y^2 - 17y + 72 = (y - 8)(y - 9) = 0 \]

### Step 4: Find the roots of the second equation

Setting each factor to zero gives us:
1. \(y - 8 = 0 \Rightarrow y = 8\)
2. \(y - 9 = 0 \Rightarrow y = 9\)

So, the solutions for \(y\) are:
\[ y = 8 \quad \text{or} \quad y = 9 \]

### Step 5: Compare the values of \(x\) and \(y\)

Now we have the following values:
- For \(x\): \(12\) or \(9\)
- For \(y\): \(8\) or \(9\)

We can compare these values:
1. If \(x = 12\), then \(12 > 8\) and \(12 > 9\).
2. If \(x = 9\), then \(9 = 9\) and \(9 > 8\).

### Conclusion

From the comparisons, we can conclude:
- In all cases, \(x\) is greater than or equal to \(y\).

Thus, the final relationship is:
\[ x \geq y \]

### Final Answer

The correct option based on the solutions is:
\[ \text{Option C: } x \geq y \]

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