In each of the following questions two equations numbered I and II are given. You have to solve both the equations and mark the appropriate option. Give answer
I `x^2-5x-84=0`
II `y^2-4y-60=0`

Options

A. if `xlty`
B. if `xgey`
C. if `xley`
D. if x=y or relationship between x and y cannot be determined.

Answer

To solve the given equations step by step, we will first solve each equation separately and then analyze the relationship between the values of \(x\) and \(y\).

### Step 1: Solve the first equation \(x^2 - 5x - 84 = 0\)

1. **Identify the coefficients**: The equation is in the standard form \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -5\), and \(c = -84\).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \(-84\) (the product \(ac\)) and add to \(-5\) (the coefficient \(b\)).
- The factors of \(-84\) that satisfy this condition are \(-12\) and \(7\) because:
- \(-12 \times 7 = -84\)
- \(-12 + 7 = -5\)
3. **Rewrite the equation**: We can rewrite the equation as:
\[ (x - 12)(x + 7) = 0 \]
4. **Set each factor to zero**:
- \(x - 12 = 0 \Rightarrow x = 12\)
- \(x + 7 = 0 \Rightarrow x = -7\)

So, the solutions for \(x\) are \(x = 12\) and \(x = -7\).

### Step 2: Solve the second equation \(y^2 - 4y - 60 = 0\)

1. **Identify the coefficients**: This equation is also in the standard form where \(a = 1\), \(b = -4\), and \(c = -60\).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \(-60\) and add to \(-4\).
- The factors of \(-60\) that satisfy this condition are \(-10\) and \(6\) because:
- \(-10 \times 6 = -60\)
- \(-10 + 6 = -4\)
3. **Rewrite the equation**: We can rewrite the equation as:
\[ (y - 10)(y + 6) = 0 \]
4. **Set each factor to zero**:
- \(y - 10 = 0 \Rightarrow y = 10\)
- \(y + 6 = 0 \Rightarrow y = -6\)

So, the solutions for \(y\) are \(y = 10\) and \(y = -6\).

### Step 3: Analyze the relationship between \(x\) and \(y\)

Now we have the following values:
- \(x\): \(12\) and \(-7\)
- \(y\): \(10\) and \(-6\)

We will compare each possible pair of \(x\) and \(y\):
1. **Pair \(x = -7\) and \(y = -6\)**:
- \(-7 < -6\) (so \(x < y\))
2. **Pair \(x = -7\) and \(y = 10\)**:
- \(-7 < 10\) (so \(x < y\))
3. **Pair \(x = 12\) and \(y = -6\)**:
- \(12 > -6\) (so \(x > y\))
4. **Pair \(x = 12\) and \(y = 10\)**:
- \(12 > 10\) (so \(x > y\))

### Conclusion

From the analysis, we see that:
- In some cases, \(x < y\) and in other cases, \(x > y\).
- Therefore, the relationship between \(x\) and \(y\) cannot be determined.

### Final Answer
The correct option is that the relationship between \(x\) and \(y\) cannot be determined.

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