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The given function \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases} 
\frac{x^2 + ax - 12}{x - 3} & \text{for } x \neq 3 \\ 
b & \text{for } x = 3 
\end{cases}
\]
We are asked to find the values of constants \( a \) and \( b \) that make \( f(x) \) continuous for all real values of \( x \).

### Step-by-Step Solution

1. **Continuity at \( x = 3 \):**  
   For \( f(x) \) to be continuous at \( x = 3 \), the following condition must be met:
   \[
   \lim_{x \to 3} f(x) = f(3) = b.
   \]
   This means that the limit of \( f(x) \) as \( x \) approaches 3 (from the form \( \frac{x^2 + ax - 12}{x - 3} \)) must equal \( b \).

2. **Simplifying the Expression for \( x \neq 3 \):**  
   Let's simplify the expression \( \frac{x^2 + ax - 12}{x - 3} \).
   We need to factor \( x^2 + ax - 12 \) so that it can be divided by \( x - 3 \):
   \[
   x^2 + ax - 12 = (x - 3)(x + c),
   \]
   where we need to determine \( c \) and \( a \) such that this factorization holds.

3. **Finding \( a \):**  
   Expanding \( (x - 3)(x + c) \) gives:
   \[
   x^2 + cx - 3x - 3c.
   \]


Find the values of constants a and b such that the function f(x) is continuous for all real numbers.

Learn how to find the constants a and b to make a piecewise function continuous at a point. This problem covers continuity, limits, and polynomial factorization, and is suitable for high school students in Grades 10-12.

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