In each of the following questions, two equations (I)and (II) are given. Solve the equations and mark the correct options
I. ` 6 x^(2) + 5x + 1 = 0 `
II. `4y ^(2) - 15 y = 4 `

Options

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

Answer

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

### Step 1: Solve the first equation \( 6x^2 + 5x + 1 = 0 \)

1. **Identify the coefficients**: Here, \( a = 6 \), \( b = 5 \), and \( c = 1 \).
2. **Use the quadratic formula**: The solutions for \( ax^2 + bx + c = 0 \) can be found using the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
3. **Calculate the discriminant**:
\[ b^2 - 4ac = 5^2 - 4 \cdot 6 \cdot 1 = 25 - 24 = 1 \]
4. **Find the roots**:
\[ x = \frac{-5 \pm \sqrt{1}}{2 \cdot 6} = \frac{-5 \pm 1}{12} \]
- First root:
\[ x_1 = \frac{-5 + 1}{12} = \frac{-4}{12} = -\frac{1}{3} \]
- Second root:
\[ x_2 = \frac{-5 - 1}{12} = \frac{-6}{12} = -\frac{1}{2} \]

### Step 2: Solve the second equation \( 4y^2 - 15y = 4 \)

1. **Rearrange the equation**:
\[ 4y^2 - 15y - 4 = 0 \]
2. **Identify the coefficients**: Here, \( a = 4 \), \( b = -15 \), and \( c = -4 \).
3. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
4. **Calculate the discriminant**:
\[ b^2 - 4ac = (-15)^2 - 4 \cdot 4 \cdot (-4) = 225 + 64 = 289 \]
5. **Find the roots**:
\[ y = \frac{15 \pm \sqrt{289}}{2 \cdot 4} = \frac{15 \pm 17}{8} \]
- First root:
\[ y_1 = \frac{32}{8} = 4 \]
- Second root:
\[ y_2 = \frac{-2}{8} = -\frac{1}{4} \]

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

1. **Values found**:
- For \( x \): \( -\frac{1}{3}, -\frac{1}{2} \)
- For \( y \): \( 4, -\frac{1}{4} \)

2. **Compare \( x \) with \( y \)**:
- For \( x = -\frac{1}{3} \):
- Compare with \( y = 4 \):
\[ -\frac{1}{3} < 4 \quad \text{(True)} \]
- Compare with \( y = -\frac{1}{4} \):
\[ -\frac{1}{3} < -\frac{1}{4} \quad \text{(False)} \]

- For \( x = -\frac{1}{2} \):
- Compare with \( y = 4 \):
\[ -\frac{1}{2} < 4 \quad \text{(True)} \]
- Compare with \( y = -\frac{1}{4} \):
\[ -\frac{1}{2} < -\frac{1}{4} \quad \text{(False)} \]

### Conclusion

From the comparisons, we find:
- Both values of \( x \) are less than \( y = 4 \).
- Only \( -\frac{1}{3} \) is less than \( -\frac{1}{4} \).

Thus, the overall relationship is:
\[ x < y \]

### Final Answer

The correct option is \( x < y \).

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