In the following two equations questions numbered (I)and (II) are given . You have to solve both equations and Give answer
I. ` x^(2) - 17 x - 84 = 0 `
II. ` y^(2) + 4y - 117 = 0`

Options

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

Answer

To solve the given equations, we will follow a systematic approach for each equation.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 17x - 84 = 0 \]

This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \). We will factor this equation.

1. **Identify coefficients**: Here, \( a = 1 \), \( b = -17 \), and \( c = -84 \).
2. **Find two numbers that multiply to \( ac \) (which is \( 1 \times -84 = -84 \)) and add to \( b \) (which is -17)**:
- The two numbers that satisfy this are \( -21 \) and \( 4 \) because:
\[ -21 \times 4 = -84 \quad \text{and} \quad -21 + 4 = -17 \]

3. **Rewrite the equation**:
\[ x^2 - 21x + 4x - 84 = 0 \]

4. **Factor by grouping**:
\[ (x^2 - 21x) + (4x - 84) = 0 \]
\[ x(x - 21) + 4(x - 21) = 0 \]
\[ (x - 21)(x + 4) = 0 \]

5. **Set each factor to zero**:
\[ x - 21 = 0 \quad \Rightarrow \quad x = 21 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 + 4y - 117 = 0 \]

1. **Identify coefficients**: Here, \( a = 1 \), \( b = 4 \), and \( c = -117 \).
2. **Find two numbers that multiply to \( ac \) (which is \( 1 \times -117 = -117 \)) and add to \( b \) (which is 4)**:
- The two numbers that satisfy this are \( 13 \) and \( -9 \) because:
\[ 13 \times -9 = -117 \quad \text{and} \quad 13 - 9 = 4 \]

3. **Rewrite the equation**:
\[ y^2 + 13y - 9y - 117 = 0 \]

4. **Factor by grouping**:
\[ (y^2 + 13y) + (-9y - 117) = 0 \]
\[ y(y + 13) - 9(y + 13) = 0 \]
\[ (y + 13)(y - 9) = 0 \]

5. **Set each factor to zero**:
\[ y + 13 = 0 \quad \Rightarrow \quad y = -13 \]
\[ y - 9 = 0 \quad \Rightarrow \quad y = 9 \]

### Summary of Solutions
From the equations, we have:
- For Equation I: \( x = 21 \) or \( x = -4 \)
- For Equation II: \( y = -13 \) or \( y = 9 \)

### Comparison of Values
Now we compare the values of \( x \) and \( y \):
- \( 21 > 9 \)
- \( 21 > -13 \)
- \( -4 < 9 \)
- \( -4 < -13 \)

### Conclusion
From the comparisons, we can conclude:
- In the case of \( x = 21 \), \( x \) is greater than both values of \( y \).
- In the case of \( x = -4 \), \( x \) is less than both values of \( y \).

Thus, there is no consistent relationship between \( x \) and \( y \) since one value of \( x \) is greater than both values of \( y \) while the other is less.