In questions two equations numbered I and II are given you have to solve both equations and give answer
`I. 25x^2+35x+12=0 II. 10y^2+9y+2=0`

Options

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

Answer

To solve the given equations, we will follow these steps:

### Step 1: Solve the first equation \( I: 25x^2 + 35x + 12 = 0 \)

1. **Identify the coefficients**:
- \( a = 25 \)
- \( b = 35 \)
- \( c = 12 \)

2. **Calculate the product \( ac \)**:
\[ ac = 25 \times 12 = 300 \]

3. **Find two numbers that multiply to \( ac \) (300) and add to \( b \) (35)**:
- The numbers are \( 20 \) and \( 15 \) because \( 20 \times 15 = 300 \) and \( 20 + 15 = 35 \).

4. **Rewrite the equation**:
\[ 25x^2 + 20x + 15x + 12 = 0 \]

5. **Factor by grouping**:
\[ 5x(5x + 4) + 3(5x + 4) = 0 \]
\[ (5x + 3)(5x + 4) = 0 \]

6. **Set each factor to zero**:
- \( 5x + 3 = 0 \) → \( x = -\frac{3}{5} \) → \( x = -0.6 \)
- \( 5x + 4 = 0 \) → \( x = -\frac{4}{5} \) → \( x = -0.8 \)

### Step 2: Solve the second equation \( II: 10y^2 + 9y + 2 = 0 \)

1. **Identify the coefficients**:
- \( a = 10 \)
- \( b = 9 \)
- \( c = 2 \)

2. **Calculate the product \( ac \)**:
\[ ac = 10 \times 2 = 20 \]

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**:
\[ 10y^2 + 5y + 4y + 2 = 0 \]

5. **Factor by grouping**:
\[ 5y(2y + 1) + 2(2y + 1) = 0 \]
\[ (5y + 2)(2y + 1) = 0 \]

6. **Set each factor to zero**:
- \( 5y + 2 = 0 \) → \( y = -\frac{2}{5} \) → \( y = -0.4 \)
- \( 2y + 1 = 0 \) → \( y = -\frac{1}{2} \) → \( y = -0.5 \)

### Step 3: Compare the values of \( x \) and \( y \)

- The values of \( x \) are \( -0.6 \) and \( -0.8 \).
- The values of \( y \) are \( -0.4 \) and \( -0.5 \).

### Step 4: Determine the relationship between \( x \) and \( y \)

1. **Compare \( x = -0.6 \) with \( y = -0.4 \)**:
- \( -0.6 < -0.4 \) → \( x \) is smaller than \( y \).

2. **Compare \( x = -0.8 \) with \( y = -0.5 \)**:
- \( -0.8 < -0.5 \) → \( x \) is smaller than \( y \).

### Conclusion

In both cases, \( x \) is smaller than \( y \).

### Final Answer
- The correct answer is: \( x < y \).