In the following questions two equations numbered I and II are given. You have to solve both the equations and answer
`x^2+12x+35=0`
`5y^2 +33y+40=0`

Options

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

Answer

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

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

1. **Identify the coefficients**: The equation is in the standard quadratic form \( ax^2 + bx + c = 0 \) where:
- \( a = 1 \)
- \( b = 12 \)
- \( c = 35 \)

2. **Factor the quadratic**: We need to find two numbers that multiply to \( c \) (35) and add up to \( b \) (12).
- The numbers are 5 and 7 because \( 5 \times 7 = 35 \) and \( 5 + 7 = 12 \).

3. **Rewrite the equation**: We can factor the equation as:
\[ (x + 5)(x + 7) = 0 \]

4. **Set each factor to zero**:
- \( x + 5 = 0 \) → \( x = -5 \)
- \( x + 7 = 0 \) → \( x = -7 \)

5. **Solutions for x**: The solutions for \( x \) are \( x = -5 \) and \( x = -7 \).

### Step 2: Solve the second equation \( 5y^2 + 33y + 40 = 0 \)

1. **Identify the coefficients**: The equation is also in the standard quadratic form \( ay^2 + by + c = 0 \) where:
- \( a = 5 \)
- \( b = 33 \)
- \( c = 40 \)

2. **Factor the quadratic**: We need to find two numbers that multiply to \( a \cdot c \) (200) and add up to \( b \) (33).
- The numbers are 25 and 8 because \( 25 \times 8 = 200 \) and \( 25 + 8 = 33 \).

3. **Rewrite the equation**: We can factor the equation as:
\[ 5y^2 + 25y + 8y + 40 = 0 \]
This can be grouped as:
\[ 5y(y + 5) + 8(y + 5) = 0 \]
Which simplifies to:
\[ (5y + 8)(y + 5) = 0 \]

4. **Set each factor to zero**:
- \( 5y + 8 = 0 \) → \( 5y = -8 \) → \( y = -\frac{8}{5} \) or \( y = -1.6 \)
- \( y + 5 = 0 \) → \( y = -5 \)

5. **Solutions for y**: The solutions for \( y \) are \( y = -\frac{8}{5} \) (or -1.6) and \( y = -5 \).

### Step 3: Compare the values of x and y

- We have \( x = -5 \) or \( x = -7 \)
- We have \( y = -1.6 \) or \( y = -5 \)

### Conclusion

- Comparing the values:
- If \( x = -5 \) and \( y = -1.6 \): \( x > y \)
- If \( x = -7 \) and \( y = -5 \): \( x < y \)

Thus, we conclude that \( x \) can be either less than or equal to \( y \).

### Final Answer

The answer is that \( x \) is either less than or equal to \( y \).

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