If `alpha, beta` are the roots of the quadratic equation `6x^(2)-6x+1=0`, then
`(1)/(2) (a+balpha+c alpha^(2)+d alpha^(3))+(1)/(2) (a+b beta+c beta^(2)+ d beta^(3))=` …..

Answer

To solve the problem, we need to analyze the given quadratic equation and use the properties of its roots. The quadratic equation provided is:

\[ 6x^2 - 6x + 1 = 0 \]

### Step 1: Identify the coefficients
The coefficients of the quadratic equation are:
- \( a = 6 \)
- \( b = -6 \)
- \( c = 1 \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{-6}{6} = 1 \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{1}{6} \)

### Step 3: Rewrite the expression
We need to evaluate the expression:

\[ \frac{1}{2} \left( a + b\alpha + c\alpha^2 + d\alpha^3 \right) + \frac{1}{2} \left( a + b\beta + c\beta^2 + d\beta^3 \right) \]

This can be simplified to:

\[ \frac{1}{2} \left( 2a + b(\alpha + \beta) + c(\alpha^2 + \beta^2) + d(\alpha^3 + \beta^3) \right) \]

### Step 4: Substitute known values
We already know \( \alpha + \beta = 1 \). Next, we need to find \( \alpha^2 + \beta^2 \) and \( \alpha^3 + \beta^3 \).

#### Finding \( \alpha^2 + \beta^2 \)
Using the identity:

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

Substituting the known values:

\[ \alpha^2 + \beta^2 = 1^2 - 2 \cdot \frac{1}{6} = 1 - \frac{1}{3} = \frac{2}{3} \]

#### Finding \( \alpha^3 + \beta^3 \)
Using the identity:

\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 + \beta^2 - \alpha\beta) \]

Substituting the known values:

\[ \alpha^3 + \beta^3 = 1 \left( \frac{2}{3} - \frac{1}{6} \right) = 1 \left( \frac{4}{6} - \frac{1}{6} \right) = 1 \cdot \frac{3}{6} = \frac{1}{2} \]

### Step 5: Substitute back into the expression
Now we can substitute \( \alpha + \beta \), \( \alpha^2 + \beta^2 \), and \( \alpha^3 + \beta^3 \) back into the expression:

\[ \frac{1}{2} \left( 2a + b(1) + c\left(\frac{2}{3}\right) + d\left(\frac{1}{2}\right) \right) \]

### Step 6: Final expression
This simplifies to:

\[ \frac{1}{2} \left( 2a + b + \frac{2c}{3} + \frac{d}{2} \right) \]

### Step 7: Final Answer
Thus, the final answer is:

\[ \frac{1}{2} \left( 2a + b + \frac{2c}{3} + \frac{d}{2} \right) \]