If `alpha, beta` be the roots of the quadratic equation `3x^(2)-6x+4=0`, find the value of
`((alpha)/(beta)+(beta)/(alpha))+2((1)/(alpha)+(1)/(beta))+3alpha beta`:

Options

A. 6
B. 8
C. 7
D. 5

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} + 2\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) + 3\alpha\beta \]

where \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(3x^2 - 6x + 4 = 0\).

### Step 1: Find the roots of the quadratic equation

The roots of the quadratic equation \(ax^2 + bx + c = 0\) can be found using the formulas:

\[ \alpha + \beta = -\frac{b}{a} \quad \text{and} \quad \alpha \beta = \frac{c}{a} \]

For the given equation \(3x^2 - 6x + 4 = 0\):

- \(a = 3\)
- \(b = -6\)
- \(c = 4\)

Calculating the sum and product of the roots:

\[ \alpha + \beta = -\frac{-6}{3} = \frac{6}{3} = 2 \]
\[ \alpha \beta = \frac{4}{3} \]

### Step 2: Calculate \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)

Using the identity:

\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} \]

We can find \(\alpha^2 + \beta^2\) using:

\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

Substituting the values we found:

\[ \alpha^2 + \beta^2 = (2)^2 - 2 \cdot \frac{4}{3} = 4 - \frac{8}{3} = \frac{12}{3} - \frac{8}{3} = \frac{4}{3} \]

Now substituting back:

\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\frac{4}{3}}{\frac{4}{3}} = 1 \]

### Step 3: Calculate \(2\left(\frac{1}{\alpha} + \frac{1}{\beta}\right)\)

Using the identity:

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \]

Substituting the values:

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{2}{\frac{4}{3}} = 2 \cdot \frac{3}{4} = \frac{3}{2} \]

Thus:

\[ 2\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) = 2 \cdot \frac{3}{2} = 3 \]

### Step 4: Calculate \(3\alpha\beta\)

We already found \(\alpha \beta = \frac{4}{3}\):

\[ 3\alpha\beta = 3 \cdot \frac{4}{3} = 4 \]

### Step 5: Combine all parts

Now we can combine all the parts we calculated:

\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} + 2\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) + 3\alpha\beta = 1 + 3 + 4 = 8 \]

### Final Answer

The value of the expression is:

\[ \boxed{8} \]