To factor the trinomial [tex]\( x^2 - 9x + 18 \)[/tex], let's follow a systematic approach:
### Step-by-Step Solution
1. Identify the coefficients:
- The given trinomial is [tex]\( x^2 - 9x + 18 \)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -9 \)[/tex], and [tex]\( c = 18 \)[/tex].
2. Calculate the product [tex]\( ac \)[/tex]:
- [tex]\( ac = 1 \times 18 = 18 \)[/tex].
3. Find two numbers whose product is [tex]\( ac \)[/tex] and whose sum is [tex]\( b \)[/tex]:
- We need two numbers that multiply to [tex]\( 18 \)[/tex] and add up to [tex]\( -9 \)[/tex].
- After careful consideration, the pair of numbers that meet both conditions is [tex]\( -3 \)[/tex] and [tex]\( -6 \)[/tex].
- Verification:
- Product: [tex]\( (-3) \times (-6) = 18 \)[/tex].
- Sum: [tex]\( -3 + (-6) = -9 \)[/tex].
4. Rewrite the middle term using the pair found:
- [tex]\( x^2 - 9x + 18 \)[/tex] can be rewritten as [tex]\( x^2 - 3x - 6x + 18 \)[/tex].
5. Factor by grouping:
- Group the terms into pairs: [tex]\( (x^2 - 3x) - (6x - 18) \)[/tex].
- Factor out the common factors in each pair:
- [tex]\( x(x - 3) - 6(x - 3) \)[/tex].
6. Factor out the common binomial factor:
- [tex]\( (x - 3) \)[/tex] is the common factor from both groups.
- Thus, we have: [tex]\( (x - 3)(x - 6) \)[/tex].
### Conclusion
The factored form of the trinomial [tex]\( x^2 - 9x + 18 \)[/tex] is:
[tex]\[ (x - 3)(x - 6) \][/tex]
