I. ` 20x^(2) + 9x+1 = 0" "` II. `30y^(2) + 11y + 1= 0`

Options

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

Explanation

To solve the given quadratic equations step by step, we will apply the factorization method for both equations.

### Step 1: Solve the first equation \( 20x^2 + 9x + 1 = 0 \)

1. **Identify coefficients**:
- \( a = 20 \)
- \( b = 9 \)
- \( c = 1 \)

2. **Multiply \( a \) and \( c \)**:
- \( ac = 20 \times 1 = 20 \)

3. **Find two numbers that multiply to \( ac \) (20) and add to \( b \) (9)**:
- The numbers are \( 5 \) and \( 4 \) because \( 5 \times 4 = 20 \) and \( 5 + 4 = 9 \).

4. **Rewrite the equation**:
- \( 20x^2 + 5x + 4x + 1 = 0 \)

5. **Factor by grouping**:
- Group the terms: \( (20x^2 + 5x) + (4x + 1) = 0 \)
- Factor out common terms: \( 5x(4x + 1) + 1(4x + 1) = 0 \)
- Combine: \( (5x + 1)(4x + 1) = 0 \)

6. **Set each factor to zero**:
- \( 5x + 1 = 0 \) or \( 4x + 1 = 0 \)

7. **Solve for \( x \)**:
- From \( 5x + 1 = 0 \): \( x = -\frac{1}{5} \)
- From \( 4x + 1 = 0 \): \( x = -\frac{1}{4} \)

### Step 2: Solve the second equation \( 30y^2 + 11y + 1 = 0 \)

1. **Identify coefficients**:
- \( a = 30 \)
- \( b = 11 \)
- \( c = 1 \)

2. **Multiply \( a \) and \( c \)**:
- \( ac = 30 \times 1 = 30 \)

3. **Find two numbers that multiply to \( ac \) (30) and add to \( b \) (11)**:
- The numbers are \( 6 \) and \( 5 \) because \( 6 \times 5 = 30 \) and \( 6 + 5 = 11 \).

4. **Rewrite the equation**:
- \( 30y^2 + 6y + 5y + 1 = 0 \)

5. **Factor by grouping**:
- Group the terms: \( (30y^2 + 6y) + (5y + 1) = 0 \)
- Factor out common terms: \( 6y(5y + 1) + 1(5y + 1) = 0 \)
- Combine: \( (6y + 1)(5y + 1) = 0 \)

6. **Set each factor to zero**:
- \( 6y + 1 = 0 \) or \( 5y + 1 = 0 \)

7. **Solve for \( y \)**:
- From \( 6y + 1 = 0 \): \( y = -\frac{1}{6} \)
- From \( 5y + 1 = 0 \): \( y = -\frac{1}{5} \)

### Summary of Solutions:
- The values of \( x \) are \( -\frac{1}{5} \) and \( -\frac{1}{4} \).
- The values of \( y \) are \( -\frac{1}{6} \) and \( -\frac{1}{5} \).