To factorize the quadratic expression [tex]\( x^2 + 10x + 21 \)[/tex], we follow these steps:
1. Identify the quadratic expression: The given expression is [tex]\( x^2 + 10x + 21 \)[/tex].
2. Find two numbers whose product is the constant term (21) and whose sum is the coefficient of the linear term (10):
- We need to determine two numbers that multiply to 21 and add up to 10.
3. Determine the factors:
- The possible pairs of factors for 21 are:
- [tex]\( 1 \times 21 \)[/tex] (which sums up to 22)
- [tex]\( 3 \times 7 \)[/tex] (which sums up to 10)
- [tex]\( -3 \times -7 \)[/tex] (which sums up to -10)
4. Select the correct pair:
- The pair that both adds up to 10 and multiplies to 21 is [tex]\( 3 \)[/tex] and [tex]\( 7 \)[/tex].
5. Write the factorization:
- Using the factors [tex]\( 3 \)[/tex] and [tex]\( 7 \)[/tex], we can write the quadratic expression [tex]\( x^2 + 10x + 21 \)[/tex] as the product of two binomials:
[tex]\[
(x + 3)(x + 7)
\][/tex]
Therefore, the factorization of [tex]\( x^2 + 10x + 21 \)[/tex] is [tex]\((x + 3)(x + 7)\)[/tex].
