In each of these questions, two equations I. and II. are given. You have to solve both the equations and give answer
I. ` x^(2) - x - 6 = 0 `
II. ` y^(2) - 6y + 8 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y`
C. If x = y or no relation can be established between x and y
D. If `y gt x `

Answer

To solve the given equations step by step, we will start with each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - x - 6 = 0 \]

This is a quadratic equation, and we can factor it. We need to find two numbers that multiply to \(-6\) (the constant term) and add up to \(-1\) (the coefficient of \(x\)).

The numbers that satisfy this are \(2\) and \(-3\). Thus, we can factor the equation as:
\[ (x - 3)(x + 2) = 0 \]

Now, we set each factor to zero:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

So, the solutions for \(x\) are:
\[ x = 3 \quad \text{and} \quad x = -2 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 6y + 8 = 0 \]

Again, this is a quadratic equation. We need to find two numbers that multiply to \(8\) (the constant term) and add up to \(-6\) (the coefficient of \(y\)).

The numbers that satisfy this are \(-2\) and \(-4\). Thus, we can factor the equation as:
\[ (y - 2)(y - 4) = 0 \]

Now, we set each factor to zero:
1. \( y - 2 = 0 \) → \( y = 2 \)
2. \( y - 4 = 0 \) → \( y = 4 \)

So, the solutions for \(y\) are:
\[ y = 2 \quad \text{and} \quad y = 4 \]

### Step 3: Compare the Values
Now we have the solutions:
- For \(x\): \(3\) and \(-2\)
- For \(y\): \(2\) and \(4\)

We can compare the values:
1. Comparing \(3\) and \(2\):
- \(3 > 2\)
2. Comparing \(3\) and \(4\):
- \(3 < 4\)
3. Comparing \(-2\) and \(2\):
- \(-2 < 2\)
4. Comparing \(-2\) and \(4\):
- \(-2 < 4\)

### Conclusion
From the comparisons:
- \(3 > 2\) and \(3 < 4\) implies that \(x\) can be greater than \(y\) in one case and less in another.
- \(-2 < 2\) and \(-2 < 4\) shows that \(-2\) is less than both values of \(y\).

Since we have conflicting results for \(x\) compared to \(y\), we conclude that no consistent relationship can be established between \(x\) and \(y\).

### Final Answer
No relation can be established between \(x\) and \(y\).

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