Sure, let's go through a detailed, step-by-step solution for the given function:
[tex]\[ \frac{x + 4}{x^2 - 5x + 6} \][/tex]
### Step 1: Factor the Denominator
First, let's factor the quadratic expression in the denominator [tex]\(x^2 - 5x + 6\)[/tex]. We need to find two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of [tex]\(x\)[/tex]).
The pairs of numbers that multiply to 6 are:
- [tex]\(1 \cdot 6\)[/tex]
- [tex]\(2 \cdot 3\)[/tex]
- [tex]\(-1 \cdot -6\)[/tex]
- [tex]\(-2 \cdot -3\)[/tex]
We need the pair that adds up to -5, which is [tex]\(-2\)[/tex] and [tex]\(-3\)[/tex]:
[tex]\[ x^2 - 5x + 6 = (x - 2)(x - 3) \][/tex]
### Step 2: Rewrite the Function Using the Factored Denominator
Now that we've factored the denominator, we can rewrite the given function:
[tex]\[ \frac{x + 4}{(x - 2)(x - 3)} \][/tex]
### Step 3: Simplify the Function if Possible
Next, we check if there's any common factor that can be canceled out from the numerator and the factored denominator. In this case, there is no common factor between [tex]\(x + 4\)[/tex] and [tex]\((x - 2)(x - 3)\)[/tex], so we cannot simplify it further.
### Step 4: Identify Any Restrictions
It's important to recognize that the values of [tex]\(x\)[/tex] which make the denominator equal to zero are not allowed, as they would make the function undefined. Thus, we set the factored denominator equal to zero to find these values:
[tex]\[ (x - 2)(x - 3) = 0 \][/tex]
[tex]\[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \][/tex]
[tex]\[ x = 2 \quad \text{or} \quad x = 3 \][/tex]
So, the function is undefined for [tex]\(x = 2\)[/tex] and [tex]\(x = 3\)[/tex].
### Conclusion
Therefore, the given function, in its simplest form, is:
[tex]\[ \frac{x + 4}{(x - 2)(x - 3)} \][/tex]
And it is important to note the restrictions:
[tex]\[ x \neq 2, \, x \neq 3 \][/tex]
### Summary
[tex]\[
\frac{x + 4}{x^2 - 5x + 6} = \frac{x + 4}{(x - 2)(x - 3)}, \quad x \neq 2, x \neq 3
\][/tex]
I hope this step-by-step explanation helps you understand how to analyze and simplify the given function! If you have any more questions, feel free to ask.
