To factor the quadratic expression [tex]\( x^2 - 2x - 15 \)[/tex], follow these steps:
1. Identify the coefficients: In the quadratic expression [tex]\( x^2 - 2x - 15 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex]),
- [tex]\( b = -2 \)[/tex] (the coefficient of [tex]\( x \)[/tex]),
- [tex]\( c = -15 \)[/tex] (the constant term).
2. Look for two numbers that multiply to [tex]\( a \cdot c \)[/tex] and add up to [tex]\( b \)[/tex]:
- Here, [tex]\( a \cdot c = 1 \cdot (-15) = -15 \)[/tex].
- We need to find two numbers whose product is -15 and whose sum is -2.
3. Find the pair of numbers:
- Consider the pairs of factors of -15:
[tex]\[
(3, -5), (-3, 5), (15, -1), (-15, 1)
\][/tex]
- From these pairs of numbers, the pair that adds up to -2 is [tex]\( 3 \)[/tex] and [tex]\(-5\)[/tex] because [tex]\( 3 + (-5) = -2 \)[/tex].
4. Rewrite the middle term using these numbers:
- Rewrite [tex]\( -2x \)[/tex] as [tex]\( 3x - 5x \)[/tex]:
[tex]\[
x^2 - 2x - 15 = x^2 + 3x - 5x - 15
\][/tex]
5. Factor by grouping:
- Group the terms in pairs:
[tex]\[
(x^2 + 3x) + (-5x - 15)
\][/tex]
- Factor out the greatest common factor (GCF) from each pair:
[tex]\[
x(x + 3) - 5(x + 3)
\][/tex]
6. Factor out the common binomial factor:
- Both groups contain the common factor [tex]\( (x + 3) \)[/tex]:
[tex]\[
(x - 5)(x + 3)
\][/tex]
So, the factored form of the expression [tex]\( x^2 - 2x - 15 \)[/tex] is:
[tex]\[
(x - 5)(x + 3)
\][/tex]
