If `x-3` is a factor of `ax^(2)-a^(2)x+12`, where a is a positive constant, what is the value of a ?

Answer

To solve the problem, we need to determine the value of \( a \) given that \( x - 3 \) is a factor of the polynomial \( ax^2 - a^2x + 12 \).

### Step-by-Step Solution:

1. **Substitute \( x = 3 \)**: Since \( x - 3 \) is a factor, substituting \( x = 3 \) into the polynomial should yield 0.
\[ a(3^2) - a^2(3) + 12 = 0 \]
Simplifying this gives:
\[ 9a - 3a^2 + 12 = 0 \]

2. **Rearranging the equation**: We can rearrange the equation to isolate the terms:
\[ -3a^2 + 9a + 12 = 0 \]
To make the leading coefficient positive, we can multiply the entire equation by -1:
\[ 3a^2 - 9a - 12 = 0 \]

3. **Divide the equation by 3**: To simplify, we can divide the entire equation by 3:
\[ a^2 - 3a - 4 = 0 \]

4. **Factoring the quadratic equation**: We can factor the quadratic equation:
\[ (a - 4)(a + 1) = 0 \]

5. **Finding the roots**: Setting each factor to zero gives us:
\[ a - 4 = 0 \quad \Rightarrow \quad a = 4 \]
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

6. **Selecting the positive value**: Since \( a \) is specified to be a positive constant, we select:
\[ a = 4 \]

### Final Answer:
The value of \( a \) is \( \boxed{4} \).