In the given questions, two equations (I) & (II) are given . You have to solve both the equations and mark the answer accordingly
I.` x^(2) + 9 x + 20 = 0`
II. ` 8 y^(2) - 15 y + 7 = 0`

Options

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

Answer

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

### Step 1: Solve Equation I: \( x^2 + 9x + 20 = 0 \)

1. **Identify the coefficients**: The equation is in the form \( ax^2 + bx + c = 0 \) where \( a = 1, b = 9, c = 20 \).
2. **Find two numbers that multiply to \( c \) (20) and add to \( b \) (9)**:
- The numbers are 4 and 5 because \( 4 \times 5 = 20 \) and \( 4 + 5 = 9 \).
3. **Rewrite the equation using these factors**:
\[ x^2 + 4x + 5x + 20 = 0 \]
4. **Group the terms**:
\[ (x^2 + 4x) + (5x + 20) = 0 \]
5. **Factor by grouping**:
\[ x(x + 4) + 5(x + 4) = 0 \]
\[ (x + 4)(x + 5) = 0 \]
6. **Set each factor to zero**:
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]
\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]

### Step 2: Solve Equation II: \( 8y^2 - 15y + 7 = 0 \)

1. **Identify the coefficients**: The equation is in the form \( ay^2 + by + c = 0 \) where \( a = 8, b = -15, c = 7 \).
2. **Find two numbers that multiply to \( a \cdot c \) (56) and add to \( b \) (-15)**:
- The numbers are -8 and -7 because \( -8 \times -7 = 56 \) and \( -8 + -7 = -15 \).
3. **Rewrite the equation using these factors**:
\[ 8y^2 - 8y - 7y + 7 = 0 \]
4. **Group the terms**:
\[ (8y^2 - 8y) + (-7y + 7) = 0 \]
5. **Factor by grouping**:
\[ 8y(y - 1) - 7(y - 1) = 0 \]
\[ (y - 1)(8y - 7) = 0 \]
6. **Set each factor to zero**:
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]
\[ 8y - 7 = 0 \quad \Rightarrow \quad y = \frac{7}{8} \]

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

- The values obtained are:
- From Equation I: \( x = -4 \) and \( x = -5 \)
- From Equation II: \( y = 1 \) and \( y = \frac{7}{8} \)

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

- Since \( -5 < -4 < \frac{7}{8} < 1 \), we conclude:
- \( x < y \)

### Final Answer
The relationship between \( x \) and \( y \) is \( x < y \).

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