Sure, let's solve the problem step-by-step.
We are given the quadratic expression:
[tex]\[ x^2 + x - 72 \][/tex]
We aim to factor this expression in the form:
[tex]\[ (x + a)(x + b) \][/tex]
### Steps to solve the problem:
1. Identify the coefficient and constant term:
- The quadratic expression is [tex]\( x^2 + x - 72 \)[/tex].
- Here, the coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 1 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\( 1 \)[/tex].
- The constant term is [tex]\( -72 \)[/tex].
2. Determine the conditions for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- We need two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \cdot b = -72 \][/tex] (product of the numbers should be equal to the constant term)
[tex]\[ a + b = 1 \][/tex] (sum of the numbers should be equal to the coefficient of [tex]\( x \)[/tex])
3. Find the correct pair [tex]\( (a, b) \)[/tex]:
- Consider the factor pairs of [tex]\(-72\)[/tex]. Some of these pairs are:
- [tex]\( (1, -72) \)[/tex]
- [tex]\( (-1, 72) \)[/tex]
- [tex]\( (2, -36) \)[/tex]
- [tex]\( (-2, 36) \)[/tex]
- [tex]\( (3, -24) \)[/tex]
- [tex]\( (-3, 24) \)[/tex]
- [tex]\( (4, -18) \)[/tex]
- [tex]\( (-4, 18) \)[/tex]
- [tex]\( (6, -12) \)[/tex]
- [tex]\( (-6, 12) \)[/tex]
- [tex]\( (8, -9) \)[/tex]
- [tex]\( (-8, 9) \)[/tex]
- From these pairs, the pair that satisfies both conditions is [tex]\( 9 \)[/tex] and [tex]\( -8 \)[/tex]:
[tex]\[ 9 \cdot (-8) = -72 \][/tex]
[tex]\[ 9 + (-8) = 1 \][/tex]
Hence, we found [tex]\( a = 9 \)[/tex] and [tex]\( b = -8 \)[/tex].
### Replace these values in the expression:
[tex]\[ (x + 9)(x - 8) \][/tex]
So, the correct answer to rewrite the given expression [tex]\( x^2 + x - 72 \)[/tex] in the form [tex]\( (x + a)(x + b) \)[/tex] is:
[tex]\[ (x + 9)(x - 8) \][/tex]
