To factor the quadratic equation [tex]\( w^2 - 11w + 18 \)[/tex], we need to find the appropriate values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
1. The product [tex]\( ab \)[/tex] equals the constant term 18: [tex]\( ab = 18 \)[/tex].
2. The sum [tex]\( a + b \)[/tex] equals the coefficient of the linear term [tex]\(-11\)[/tex]: [tex]\( a + b = 11 \)[/tex].
We need to identify two numbers that satisfy these criteria.
Let's list the pairs of factors of 18:
[tex]\[ (1, 18), (2, 9), (3, 6), (-1, -18), (-2, -9), (-3, -6) \][/tex]
Now, we look for the pair of factors whose sum is 11:
- [tex]\( 1 + 18 = 19 \)[/tex]
- [tex]\( 2 + 9 = 11 \)[/tex]
- [tex]\( 3 + 6 = 9 \)[/tex]
- [tex]\( -1 + (-18) = -19 \)[/tex]
- [tex]\( -2 + (-9) = -11 \)[/tex]
- [tex]\( -3 + (-6) = -9 \)[/tex]
From these pairs, we can see that the pair:
[tex]\[ a = 2 \; \text{and} \; b = 9 \][/tex]
satisfies both conditions:
- [tex]\( 2 \times 9 = 18 \)[/tex]
- [tex]\( 2 + 9 = 11 \)[/tex]
Therefore, the quadratic [tex]\( w^2 - 11w + 18 \)[/tex] can be factored as:
[tex]\[ w^2 - 11w + 18 = (w - 2)(w - 9) \][/tex]
Hence, the correct values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[ a = 2 \; \checkmark \][/tex]
[tex]\[ b = 9 \; \square \][/tex]
