To factor the trinomial [tex]\( x^2 - 10x + 9 \)[/tex], we follow these steps:
1. Identify the coefficients from the quadratic equation [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -10 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = 9 \)[/tex] (constant term)
2. Find two numbers that multiply to [tex]\( ac \)[/tex] (product of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]) and add up to [tex]\( b \)[/tex]:
- Here, [tex]\( ac = 1 \cdot 9 = 9 \)[/tex]
- We need two numbers that multiply to 9 and add up to -10.
3. The pair of numbers that satisfies these conditions is [tex]\( -1 \)[/tex] and [tex]\( -9 \)[/tex], because:
- [tex]\( (-1) \cdot (-9) = 9 \)[/tex] (they multiply to [tex]\( c \)[/tex])
- [tex]\( -1 + (-9) = -10 \)[/tex] (they add up to [tex]\( b \)[/tex])
4. Rewrite the middle term (-10x) using the two numbers found:
[tex]\[
x^2 - 10x + 9 = x^2 - x - 9x + 9
\][/tex]
5. Factor by grouping:
- Group the terms: [tex]\((x^2 - x)\)[/tex] and [tex]\((-9x + 9)\)[/tex]
- Factor out the greatest common factor (GCF) from each group:
[tex]\[
x(x - 1) - 9(x - 1)
\][/tex]
6. Factor out the common binomial factor:
- Notice that [tex]\((x - 1)\)[/tex] is a common factor in both groups:
[tex]\[
(x - 1)(x - 9)
\][/tex]
Thus, the factored form of the trinomial [tex]\( x^2 - 10x + 9 \)[/tex] is:
[tex]\[
(x - 1)(x - 9)
\][/tex]
