In each of the following questions, two equations (I)and (II) are given. Solve the equations and mark the correct options
I. ` x^(2) - 18 x + 45 = 0 `
II. `y^(2) + 12 y - 45 = 0 `

Options

A. A)If ` x gt y `
B. B)If ` x ge y `
C. C)if ` x lt y `
D. D)If ` x le y`

Answer

To solve the given equations and determine the relationship between the values of \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation
The first equation is:
\[ x^2 - 18x + 45 = 0 \]
We will factor this quadratic equation. We need two numbers that multiply to \( 45 \) and add up to \( -18 \). The numbers are \( -15 \) and \( -3 \).

Thus, we can rewrite the equation as:
\[ (x - 15)(x - 3) = 0 \]

### Step 2: Find the values of \( x \)
Setting each factor to zero gives us:
1. \( x - 15 = 0 \) → \( x = 15 \)
2. \( x - 3 = 0 \) → \( x = 3 \)

So, the values of \( x \) are \( 15 \) and \( 3 \).

### Step 3: Solve the second equation
The second equation is:
\[ y^2 + 12y - 45 = 0 \]
We will factor this quadratic equation as well. We need two numbers that multiply to \( -45 \) and add up to \( 12 \). The numbers are \( 15 \) and \( -3 \).

Thus, we can rewrite the equation as:
\[ (y + 15)(y - 3) = 0 \]

### Step 4: Find the values of \( y \)
Setting each factor to zero gives us:
1. \( y + 15 = 0 \) → \( y = -15 \)
2. \( y - 3 = 0 \) → \( y = 3 \)

So, the values of \( y \) are \( -15 \) and \( 3 \).

### Step 5: Compare values of \( x \) and \( y \)
Now we have the values:
- For \( x \): \( 15 \) and \( 3 \)
- For \( y \): \( -15 \) and \( 3 \)

We can compare these values:
1. When \( x = 15 \), \( y = -15 \): Here, \( x > y \).
2. When \( x = 3 \), \( y = 3 \): Here, \( x = y \).

### Conclusion
From the comparisons, we can conclude:
- In one case, \( x \) is greater than \( y \) (when \( x = 15 \)).
- In the other case, \( x \) is equal to \( y \) (when \( x = 3 \)).

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

### Final Answer
The correct option is \( x \geq y \).

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