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The problem states that a quadratic function \( f(x) = ax^2 + bx + c \) has real and distinct roots. Given that \( b^2 - 4ac = 9 \) and \( a = 1 \), we need to determine the values of \( b \) and \( c \) from the provided options.

### Solution:
Since \( a = 1 \), the discriminant formula simplifies to:
\[
b^2 - 4c = 9
\]
We can substitute each option to check if they satisfy this equation.

Let's go through each choice:

1. **\( b = 3 \) and \( c = -2 \)**:
   \[
   b^2 - 4c = 3^2 - 4(-2) = 9 + 8 = 17 \neq 9
   \]
   This option is incorrect.

2. **\( b = 4 \) and \( c = 1 \)**:
   \[
   b^2 - 4c = 4^2 - 4 \cdot 1 = 16 - 4 = 12 \neq 9
   \]
   This option is incorrect.

3. **\( b = 3 \) and \( c = -1 \)**:
   \[
   b^2 - 4c = 3^2 - 4(-1) = 9 + 4 = 13 \neq 9
   \]
   This option is incorrect.

4. **\( b = 2 \) and \( c = 1 \)**:
   \[
   b^2 - 4c = 2^2 - 4 \cdot 1 = 4 - 4 = 0 \neq 9
   \]
   This option is incorrect.

5. **\( b = 3 \) and \( c = 2 \)**:
   \[
   b^2 - 4c = 3^2 - 4 \cdot 2 = 9 - 8 = 1 \neq 9
   \]
   This option is incorrect.

It appears none of the options satisfy the condition \( b^2 - 4c = 9 \). Let me know if you have further questions, or if you'd like to discuss potential follow-up questions related to discriminants and quadratic equations.

Diketahui fungsi kuadrat f(x) = ax^2 + bx + c memiliki akar-akar real dan berbeda. Jika b^2 - 4ac = 9 dan a = 1, tentukan nilai b dan c yang mungkin.

Explore how to determine the possible values of b and c for the quadratic function f(x) = ax^2 + bx + c with real and distinct roots, given the discriminant b^2 - 4ac = 9 and a = 1. This problem covers algebraic concepts and the properties of quadratic equations.

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