Solve the given quadratic equations and mark the correct options based on your answer
I.` 6 x ^(2) - 5 x + 1 = 0`
II. ` 3 y^(2) + 8 y = 3 `

Options

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

Answer

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

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

We will use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \(6x^2 - 5x + 1 = 0\), we identify \(a = 6\), \(b = -5\), and \(c = 1\).

#### Calculate the discriminant \(D\):

\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 6 \cdot 1 = 25 - 24 = 1 \]

#### Calculate the roots using the quadratic formula:

\[ x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot 6} = \frac{5 \pm 1}{12} \]

This gives us two possible values for \(x\):

1. \(x_1 = \frac{5 + 1}{12} = \frac{6}{12} = \frac{1}{2}\)
2. \(x_2 = \frac{5 - 1}{12} = \frac{4}{12} = \frac{1}{3}\)

### Step 2: Solve the second quadratic equation \(3y^2 + 8y - 3 = 0\)

Again, we will use the quadratic formula. Here, \(a = 3\), \(b = 8\), and \(c = -3\).

#### Calculate the discriminant \(D\):

\[ D = b^2 - 4ac = (8)^2 - 4 \cdot 3 \cdot (-3) = 64 + 36 = 100 \]

#### Calculate the roots using the quadratic formula:

\[ y = \frac{-8 \pm \sqrt{100}}{2 \cdot 3} = \frac{-8 \pm 10}{6} \]

This gives us two possible values for \(y\):

1. \(y_1 = \frac{-8 + 10}{6} = \frac{2}{6} = \frac{1}{3}\)
2. \(y_2 = \frac{-8 - 10}{6} = \frac{-18}{6} = -3\)

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

We have the following values:

- From the first equation: \(x_1 = \frac{1}{2}\), \(x_2 = \frac{1}{3}\)
- From the second equation: \(y_1 = \frac{1}{3}\), \(y_2 = -3\)

#### Compare \(x\) and \(y\):

1. For \(x_1 = \frac{1}{2}\) and \(y_2 = -3\):
\[ \frac{1}{2} > -3 \quad \text{(True)} \]

2. For \(x_2 = \frac{1}{3}\) and \(y_1 = \frac{1}{3}\):
\[ \frac{1}{3} = \frac{1}{3} \quad \text{(True)} \]

### Conclusion:

From the comparisons, we can conclude that:

\[ x \geq y \]

### Final Answer:

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