Quadratic Equation:
The given quadratic equation is: (a^2 - 5a + 3)x^2 + (3a - 1)x + 2 = 0.

Roots of a Quadratic Equation:
A quadratic equation ax^2 + bx + c = 0 has two roots, given by the formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In the given equation, the coefficient of x^2 is (a^2 - 5a + 3), the coefficient of x is (3a - 1), and the constant term is 2.

Conditions for the Roots:
For the roots of a quadratic equation to be real and distinct, the discriminant (b^2 - 4ac) must be positive. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Using the Discriminant:
Let's use the discriminant to find the conditions for the roots of the given equation.

Discriminant = (3a - 1)^2 - 4(a^2 - 5a + 3)(2)
= 9a^2 - 6a + 1 - 8a^2 + 40a - 24
= a^2 + 34a - 23

For the roots to be real and distinct, the discriminant must be positive.

a^2 + 34a - 23 > 0

Finding the Roots:
Now, let's find the roots of the given quadratic equation.

Using the quadratic formula, the roots are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Substituting the values from the given equation, we have:

x = (-(3a - 1) ± √((3a - 1)^2 - 4(a^2 - 5a + 3)(2))) / 2(a^2 - 5a + 3)

Simplifying further, we get:

x = (1 - 3a ± √(9a^2 - 6a + 1 - 16a^2 + 80a - 48)) / 2(a^2 - 5a + 3)
x = (1 - 3a ± √(-7a^2 + 74a - 47)) / 2(a^2 - 5a + 3)

Condition for Roots:
Since the given quadratic equation has one root twice as large as the other, we can set up the following condition:

(1 - 3a + √(-7a^2 + 74a - 47)) / (1 - 3a - √(-7a^2 + 74a - 47)) = 2

Solving the Condition:
Let's solve the above condition to find the value of a.

Multiplying both sides of the equation by (1 - 3a - √(-7a^2 + 74a - 47)), we get:

1 - 3a + √(-7a^2 + 74a - 47