In each of the following questions, two equations are given. You have to solve them and
I. `x^(2) + x - 2 = 0` II. `y^(2) - 9y + 20 = 0`

Options

A. if `x gt y`
B. if `x ge y`
C. if `x lt y`
D. if `x le y`

Answer

To solve the given equations step by step, we will follow these procedures:

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

1. We need to factor the quadratic equation. We look for two numbers that multiply to \(-2\) (the constant term) and add up to \(1\) (the coefficient of \(x\)).
2. The numbers that satisfy this condition are \(2\) and \(-1\).
3. We can rewrite the equation as:
\[ (x - 1)(x + 2) = 0 \]
4. Setting each factor to zero gives us the solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
5. Thus, the solutions for \(x\) are \(x = 1\) and \(x = -2\).

### Step 2: Solve the second equation \( y^2 - 9y + 20 = 0 \)

1. Similarly, we need to factor this quadratic equation. We look for two numbers that multiply to \(20\) and add up to \(-9\).
2. The numbers that satisfy this condition are \(-5\) and \(-4\).
3. We can rewrite the equation as:
\[ (y - 5)(y - 4) = 0 \]
4. Setting each factor to zero gives us the solutions:
\[ y - 5 = 0 \quad \Rightarrow \quad y = 5 \]
\[ y - 4 = 0 \quad \Rightarrow \quad y = 4 \]
5. Thus, the solutions for \(y\) are \(y = 5\) and \(y = 4\).

### Step 3: Compare the solutions

1. We have the values of \(x\) as \(1\) and \(-2\).
2. We have the values of \(y\) as \(5\) and \(4\).
3. Now we compare:
- For \(x = 1\), both \(y = 4\) and \(y = 5\) are greater than \(1\).
- For \(x = -2\), both \(y = 4\) and \(y = 5\) are also greater than \(-2\).

### Conclusion

From our comparisons, we can conclude that \(y\) is always greater than \(x\). Therefore, the correct relationship is:

\[ y > x \]

### Final Answer

The answer is that \(y\) is greater than \(x\).

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