Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\frac{x - 2}{2} \quad \text{where} \quad x \neq 3
\]
Solution
Sure, let's simplify the given rational expression step by step.
**Given Expression:**
\[
\frac{x^{2} - 5x + 6}{2x - 6}
\]
**Step 1: Factor the Numerator and the Denominator**
1. **Factor the numerator \( x^{2} - 5x + 6 \):**
We look for two numbers that multiply to **6** (the constant term) and add up to **-5** (the coefficient of \( x \)).
- The numbers are **-2** and **-3** because:
\[
(-2) \times (-3) = 6 \quad \text{and} \quad (-2) + (-3) = -5
\]
So, the numerator factors to:
\[
x^{2} - 5x + 6 = (x - 2)(x - 3)
\]
2. **Factor the denominator \( 2x - 6 \):**
Factor out the greatest common factor, which is **2**:
\[
2x - 6 = 2(x - 3)
\]
**Step 2: Rewrite the Expression with Factored Forms**
\[
\frac{(x - 2)(x - 3)}{2(x - 3)}
\]
**Step 3: Simplify by Cancelling Common Factors**
Notice that \( (x - 3) \) appears in both the numerator and the denominator. Assuming \( x \neq 3 \) (to avoid division by zero), we can cancel this term:
\[
\frac{(x - 2)\cancel{(x - 3)}}{2\cancel{(x - 3)}} = \frac{x - 2}{2}
\]
**Final Simplified Expression:**
\[
\frac{x - 2}{2} \quad \text{where} \quad x \neq 3
\]
**Summary:**
\[
\frac{x^{2} - 5x + 6}{2x - 6} = \frac{x - 2}{2} \quad \text{for} \quad x \neq 3
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To simplify the expression \( \frac{x^{2}-5x+6}{2x-6} \), first factor both the numerator and the denominator. The numerator \( x^2 - 5x + 6 \) can be factored into \( (x - 2)(x - 3) \), and the denominator \( 2x - 6 \) can be factored into \( 2(x - 3) \). This gives us: \[ \frac{(x - 2)(x - 3)}{2(x - 3)} \] Now, we can cancel out the common factor \( (x - 3) \) (as long as \( x \neq 3 \)), resulting in: \[ \frac{x - 2}{2} \] So, the simplified form is: \[ \frac{x - 2}{2} \quad (x \neq 3) \] This simplification helps to avoid any complications when substituting values later on.
