To factor the quadratic expression [tex]\( x^2 - 3x - 10 \)[/tex] completely, we can follow these steps:
1. Identify the quadratic expression:
We start with the expression [tex]\( x^2 - 3x - 10 \)[/tex].
2. Find the factors of the constant term (-10) that add up to the coefficient of the linear term (-3):
We need two numbers that multiply to give [tex]\(-10\)[/tex] (the constant term) and add up to [tex]\(-3\)[/tex] (the coefficient of the linear term).
These two numbers are [tex]\(-5\)[/tex] and [tex]\(2\)[/tex] because:
- [tex]\(-5 \times 2 = -10\)[/tex]
- [tex]\(-5 + 2 = -3\)[/tex]
3. Rewrite the middle term (-3x) using the two numbers found:
Thus, we can rewrite the expression as:
[tex]\[
x^2 - 3x - 10 = x^2 - 5x + 2x - 10
\][/tex]
4. Group terms to factor by grouping:
Next, we can group the terms as follows:
[tex]\[
(x^2 - 5x) + (2x - 10)
\][/tex]
5. Factor out the greatest common factor (GCF) from each group:
- From the first group ([tex]\(x^2 - 5x\)[/tex]), the GCF is [tex]\(x\)[/tex]:
[tex]\[
x(x - 5)
\][/tex]
- From the second group ([tex]\(2x - 10\)[/tex]), the GCF is [tex]\(2\)[/tex]:
[tex]\[
2(x - 5)
\][/tex]
6. Factor out the common binomial factor [tex]\((x - 5)\)[/tex]:
[tex]\[
(x - 5)(x + 2)
\][/tex]
Thus, the factored form of the quadratic expression [tex]\( x^2 - 3x - 10 \)[/tex] is:
[tex]\[
(x - 5)(x + 2)
\][/tex]
