In the following questions two equations I. and II. are given. You have to solve both the equations and mark the appropriate options
I. ` 32 x^(2) + 44 x + 15 = 0 `
II. ` 42 y^(2) + 53 y + 15 = 0 `

Options

A. ` x lt y `
B. ` x gt y `
C. ` x le y `
D. x = y or no relation

Answer

To solve the given equations and find the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation \( 32x^2 + 44x + 15 = 0 \)

We will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 32 \), \( b = 44 \), and \( c = 15 \).

#### Calculation:
1. Calculate \( b^2 - 4ac \):
\[ b^2 = 44^2 = 1936 \]
\[ 4ac = 4 \times 32 \times 15 = 1920 \]
\[ b^2 - 4ac = 1936 - 1920 = 16 \]

2. Substitute values into the quadratic formula:
\[ x = \frac{-44 \pm \sqrt{16}}{2 \times 32} \]
\[ x = \frac{-44 \pm 4}{64} \]

3. Calculate the two possible values of \( x \):
- First value:
\[ x_1 = \frac{-44 + 4}{64} = \frac{-40}{64} = -0.625 \]
- Second value:
\[ x_2 = \frac{-44 - 4}{64} = \frac{-48}{64} = -0.75 \]

### Step 2: Solve the second equation \( 42y^2 + 53y + 15 = 0 \)

Again, we will use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 42 \), \( b = 53 \), and \( c = 15 \).

#### Calculation:
1. Calculate \( b^2 - 4ac \):
\[ b^2 = 53^2 = 2809 \]
\[ 4ac = 4 \times 42 \times 15 = 2520 \]
\[ b^2 - 4ac = 2809 - 2520 = 289 \]

2. Substitute values into the quadratic formula:
\[ y = \frac{-53 \pm \sqrt{289}}{2 \times 42} \]
\[ y = \frac{-53 \pm 17}{84} \]

3. Calculate the two possible values of \( y \):
- First value:
\[ y_1 = \frac{-53 + 17}{84} = \frac{-36}{84} = -0.42857 \approx -0.4285 \]
- Second value:
\[ y_2 = \frac{-53 - 17}{84} = \frac{-70}{84} = -0.8333 \approx -0.833 \]

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

We have the following values:
- \( x_1 = -0.625 \)
- \( x_2 = -0.75 \)
- \( y_1 = -0.4285 \)
- \( y_2 = -0.833 \)

#### Comparison:
1. Compare \( y_1 \) with \( x_1 \) and \( x_2 \):
- \( y_1 = -0.4285 \) is greater than \( x_1 = -0.625 \)
- \( y_1 = -0.4285 \) is greater than \( x_2 = -0.75 \)

2. Compare \( y_2 \) with \( x_1 \) and \( x_2 \):
- \( y_2 = -0.833 \) is less than \( x_1 = -0.625 \)
- \( y_2 = -0.833 \) is less than \( x_2 = -0.75 \)

### Conclusion:
The values of \( x \) and \( y \) do not have a consistent relationship. Thus, there is no relation between \( x \) and \( y \).

### Final Answer:
**Option D: No relation between \( x \) and \( y \)**

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