Solve the following quadratic equation and mark and answer as per instructions.
I. ` x^(2) - 2 x - 143 = 0`
II. ` y^(2) - 169= 0`

Options

A. ` x gt y`
B. ` x lt y`
C. ` x le y`
D. x = y or no relation can be established

Answer

To solve the given quadratic equations step by step, we will follow the standard methods for solving quadratic equations.

### Step 1: Solve the first equation \( x^2 - 2x - 143 = 0 \)

We will factor the quadratic equation. We need to find two numbers that multiply to \(-143\) (the constant term) and add to \(-2\) (the coefficient of \(x\)).

The factors of \(-143\) that satisfy this condition are \(11\) and \(-13\) because:
- \(11 \times (-13) = -143\)
- \(11 + (-13) = -2\)

Thus, we can factor the equation as:
\[ (x - 13)(x + 11) = 0 \]

### Step 2: Set each factor to zero

Now, we set each factor equal to zero to find the values of \(x\):
1. \(x - 13 = 0 \Rightarrow x = 13\)
2. \(x + 11 = 0 \Rightarrow x = -11\)

So, the solutions for \(x\) are:
\[ x = 13 \quad \text{and} \quad x = -11 \]

### Step 3: Solve the second equation \( y^2 - 169 = 0 \)

This is a difference of squares, which can be factored as:
\[ (y - 13)(y + 13) = 0 \]

### Step 4: Set each factor to zero

Now, we set each factor equal to zero to find the values of \(y\):
1. \(y - 13 = 0 \Rightarrow y = 13\)
2. \(y + 13 = 0 \Rightarrow y = -13\)

So, the solutions for \(y\) are:
\[ y = 13 \quad \text{and} \quad y = -13 \]

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

Now we have the values:
- For \(x\): \(13\) and \(-11\)
- For \(y\): \(13\) and \(-13\)

We will compare these values:
1. Comparing \(x = 13\) with \(y = 13\):
- \(x = y\)

2. Comparing \(x = 13\) with \(y = -13\):
- \(13 > -13\) (thus, \(x > y\))

3. Comparing \(x = -11\) with \(y = 13\):
- \(-11 < 13\) (thus, \(x < y\))

4. Comparing \(x = -11\) with \(y = -13\):
- \(-11 > -13\) (thus, \(x > y\))

### Summary of Comparisons:
- \(x = 13\) is equal to \(y = 13\)
- \(x = 13\) is greater than \(y = -13\)
- \(x = -11\) is less than \(y = 13\)
- \(x = -11\) is greater than \(y = -13\)

### Conclusion:
From the comparisons, we can conclude that there is no consistent relationship between \(x\) and \(y\) across all values. Therefore, we can state that no definitive relation can be established between \(x\) and \(y\).