In the following questions two equations numbered I and II are given. Solve both the equations and give answer
I. `x^2-11x+30=0` II. `y^2-y-20=0`

Options

A. `xgey`
B. `xlty`
C. `xley`
D. relationship can't be established

Answer

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

### Step 1: Solve the first equation \( I: x^2 - 11x + 30 = 0 \)

1. **Identify the equation**: The equation is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \).
- Here, \( a = 1 \), \( b = -11 \), and \( c = 30 \).

2. **Factor the quadratic**: We need to find two numbers that multiply to \( c \) (30) and add up to \( b \) (-11).
- The numbers that satisfy this condition are -5 and -6, since:
- \(-5 \times -6 = 30\)
- \(-5 + -6 = -11\)

3. **Write the factored form**: The equation can be factored as:
\[ (x - 5)(x - 6) = 0 \]

4. **Set each factor to zero**:
- \( x - 5 = 0 \) → \( x = 5 \)
- \( x - 6 = 0 \) → \( x = 6 \)

5. **Solutions for \( x \)**: The solutions are \( x = 5 \) and \( x = 6 \).

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

1. **Identify the equation**: The equation is also a quadratic equation in the standard form \( ay^2 + by + c = 0 \).
- Here, \( a = 1 \), \( b = -1 \), and \( c = -20 \).

2. **Factor the quadratic**: We need to find two numbers that multiply to \( c \) (-20) and add up to \( b \) (-1).
- The numbers that satisfy this condition are 4 and -5, since:
- \(4 \times -5 = -20\)
- \(4 + -5 = -1\)

3. **Write the factored form**: The equation can be factored as:
\[ (y - 5)(y + 4) = 0 \]

4. **Set each factor to zero**:
- \( y - 5 = 0 \) → \( y = 5 \)
- \( y + 4 = 0 \) → \( y = -4 \)

5. **Solutions for \( y \)**: The solutions are \( y = 5 \) and \( y = -4 \).

### Step 3: Compare the solutions

- We have the solutions:
- For \( x \): \( 5, 6 \)
- For \( y \): \( 5, -4 \)

- We need to determine the relationship between \( x \) and \( y \):
- The common value is \( 5 \).
- The maximum value of \( x \) is \( 6 \) which is greater than \( y = -4 \).

### Final Answer

Since \( x \) can be \( 5 \) or \( 6 \) and \( y \) can be \( 5 \) or \( -4 \), we conclude that \( x \) is greater than or equal to \( y \).

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