In each questions two equations numbered I. and II. are given. You have to solve both the equations and mark appropriate answer
I. ` x^(2) - 2x - 24 = 0 `
II. ` y^(2) + 3y - 40 = 0`

Options

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

Answer

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

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 2x - 24 = 0 \]
To solve this quadratic equation, we can factor it. We need two numbers that multiply to \(-24\) (the constant term) and add up to \(-2\) (the coefficient of \(x\)).

The numbers that satisfy these conditions are \(6\) and \(-4\) because:
\[ 6 \times (-4) = -24 \quad \text{and} \quad 6 + (-4) = 2 \]

Now we can rewrite the equation:
\[ x^2 - 6x + 4x - 24 = 0 \]
Grouping the terms:
\[ (x - 6)(x + 4) = 0 \]
Setting each factor to zero gives us:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

So, the solutions for \(x\) are:
\[ x = 6 \quad \text{or} \quad x = -4 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 + 3y - 40 = 0 \]
Again, we will factor this quadratic equation. We need two numbers that multiply to \(-40\) (the constant term) and add up to \(3\) (the coefficient of \(y\)).

The numbers that satisfy these conditions are \(8\) and \(-5\) because:
\[ 8 \times (-5) = -40 \quad \text{and} \quad 8 + (-5) = 3 \]

Now we can rewrite the equation:
\[ y^2 + 8y - 5y - 40 = 0 \]
Grouping the terms:
\[ (y + 8)(y - 5) = 0 \]
Setting each factor to zero gives us:
\[ y + 8 = 0 \quad \Rightarrow \quad y = -8 \]
\[ y - 5 = 0 \quad \Rightarrow \quad y = 5 \]

So, the solutions for \(y\) are:
\[ y = -8 \quad \text{or} \quad y = 5 \]

### Step 3: Compare the Values
Now we have the values for \(x\) and \(y\):
- \(x = 6\) or \(x = -4\)
- \(y = -8\) or \(y = 5\)

We need to compare these values:
1. If \(x = 6\):
- Compare with \(y = -8\): \(6 > -8\)
- Compare with \(y = 5\): \(6 > 5\)

2. If \(x = -4\):
- Compare with \(y = -8\): \(-4 > -8\)
- Compare with \(y = 5\): \(-4 < 5\)

### Conclusion
From the comparisons, we can see:
- When \(x = 6\), \(x\) is greater than both values of \(y\).
- When \(x = -4\), \(x\) is greater than \(y = -8\) but less than \(y = 5\).

Thus, we cannot establish a consistent relationship between \(x\) and \(y\) as it varies based on the values chosen. Therefore, the answer is:

**No relation can be established between \(x\) and \(y\).**