(i) `x^(2)+9x+20=0`
(ii) `y^(2)+7y+12=0`

Options

A. `xgey`
B. `xgty`
C. `xley`
D. `xlty`

Answer

To solve the given equations step by step, we will follow the method of factoring.

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

1. **Identify the coefficients**: Here, the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 9 \) (coefficient of \( x \))
- \( c = 20 \) (constant term)

2. **Find two numbers that multiply to \( c \) (20) and add to \( b \) (9)**:
- The two numbers that satisfy this condition are \( 5 \) and \( 4 \) because \( 5 \times 4 = 20 \) and \( 5 + 4 = 9 \).

3. **Rewrite the equation using these factors**:
\[ x^2 + 5x + 4x + 20 = 0 \]
This can be grouped as:
\[ (x^2 + 5x) + (4x + 20) = 0 \]

4. **Factor by grouping**:
\[ x(x + 5) + 4(x + 5) = 0 \]
Now, factor out the common term \( (x + 5) \):
\[ (x + 5)(x + 4) = 0 \]

5. **Set each factor to zero**:
\[ x + 5 = 0 \quad \text{or} \quad x + 4 = 0 \]
Solving these gives:
\[ x = -5 \quad \text{or} \quad x = -4 \]

### Step 2: Solve the second equation \( y^2 + 7y + 12 = 0 \)

1. **Identify the coefficients**: Here, the coefficients are:
- \( a = 1 \)
- \( b = 7 \)
- \( c = 12 \)

2. **Find two numbers that multiply to \( c \) (12) and add to \( b \) (7)**:
- The two numbers that satisfy this condition are \( 4 \) and \( 3 \) because \( 4 \times 3 = 12 \) and \( 4 + 3 = 7 \).

3. **Rewrite the equation using these factors**:
\[ y^2 + 4y + 3y + 12 = 0 \]
This can be grouped as:
\[ (y^2 + 4y) + (3y + 12) = 0 \]

4. **Factor by grouping**:
\[ y(y + 4) + 3(y + 4) = 0 \]
Now, factor out the common term \( (y + 4) \):
\[ (y + 4)(y + 3) = 0 \]

5. **Set each factor to zero**:
\[ y + 4 = 0 \quad \text{or} \quad y + 3 = 0 \]
Solving these gives:
\[ y = -4 \quad \text{or} \quad y = -3 \]

### Final Answers:
- The solutions for \( x \) are \( -5 \) and \( -4 \).
- The solutions for \( y \) are \( -4 \) and \( -3 \).

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