Find the value of a for which one root of the quadratic equation `(a^(2)-5a+3)x^(2)+(3a-1)` x+2=0 is twice as large as the other.

Answer

To find the value of \( a \) for which one root of the quadratic equation

\[ (a^2 - 5a + 3)x^2 + (3a - 1)x + 2 = 0 \]

is twice as large as the other root, we can follow these steps:

### Step 1: Define the Roots
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). According to the problem, we have:

\[ \beta = 2\alpha \]

### Step 2: Sum of the Roots
The sum of the roots \( \alpha + \beta \) can be expressed using Vieta's formulas:

\[ \alpha + \beta = -\frac{b}{a} \]

Here, \( a = a^2 - 5a + 3 \) and \( b = 3a - 1 \). Therefore, we can write:

\[ \alpha + 2\alpha = -\frac{3a - 1}{a^2 - 5a + 3} \]

This simplifies to:

\[ 3\alpha = -\frac{3a - 1}{a^2 - 5a + 3} \]

From this, we can express \( \alpha \):

\[ \alpha = -\frac{3a - 1}{3(a^2 - 5a + 3)} \]

### Step 3: Product of the Roots
The product of the roots \( \alpha \cdot \beta \) can also be expressed using Vieta's formulas:

\[ \alpha \cdot \beta = \frac{c}{a} \]

where \( c = 2 \). Thus, we have:

\[ \alpha \cdot (2\alpha) = \frac{2}{a^2 - 5a + 3} \]

This simplifies to:

\[ 2\alpha^2 = \frac{2}{a^2 - 5a + 3} \]

From this, we can express \( \alpha^2 \):

\[ \alpha^2 = \frac{1}{a^2 - 5a + 3} \]

### Step 4: Substitute \( \alpha \)
Now we substitute the expression for \( \alpha \) into the equation for \( \alpha^2 \):

\[ \left(-\frac{3a - 1}{3(a^2 - 5a + 3)}\right)^2 = \frac{1}{a^2 - 5a + 3} \]

### Step 5: Simplify the Equation
Squaring the left side gives:

\[ \frac{(3a - 1)^2}{9(a^2 - 5a + 3)^2} = \frac{1}{a^2 - 5a + 3} \]

Cross-multiplying yields:

\[ (3a - 1)^2 = 9(a^2 - 5a + 3) \]

### Step 6: Expand and Rearrange
Expanding both sides:

\[ 9a^2 - 6a + 1 = 9a^2 - 45a + 27 \]

Subtracting \( 9a^2 \) from both sides:

\[ -6a + 1 = -45a + 27 \]

Rearranging gives:

\[ 39a = 26 \]

### Step 7: Solve for \( a \)
Dividing both sides by 39:

\[ a = \frac{26}{39} = \frac{2}{3} \]

### Final Answer
The value of \( a \) is:

\[ \boxed{\frac{2}{3}} \]