To factor the quadratic expression [tex]\( 21x^2 + 6x \)[/tex] completely, follow these detailed steps:
1. Identify the common factor:
- Both terms in the expression [tex]\( 21x^2 \)[/tex] and [tex]\( 6x \)[/tex] have a common factor, which is the greatest common factor (GCF). In this case, the GCF of 21 and 6 is 3.
- Both terms also have the variable [tex]\( x \)[/tex] in common.
2. Factor out the greatest common factor:
- We can factor out [tex]\( 3x \)[/tex] from each term in the expression.
- When we factor out [tex]\( 3x \)[/tex], we essentially divide each term by [tex]\( 3x \)[/tex].
So,
[tex]\[
21x^2 + 6x = 3x \cdot \left( \frac{21x^2}{3x} \right) + 3x \cdot \left( \frac{6x}{3x} \right)
\][/tex]
Simplifying inside the parentheses:
[tex]\[
21x^2 + 6x = 3x \cdot (7x) + 3x \cdot (2)
\][/tex]
3. Write the factored expression:
- Combine the factored terms to get the final factored expression.
[tex]\[
21x^2 + 6x = 3x(7x + 2)
\][/tex]
Thus, the completely factored form of the quadratic expression [tex]\( 21x^2 + 6x \)[/tex] is:
[tex]\[
3x(7x + 2)
\][/tex]
### Conclusion
The quadratic expression [tex]\( 21x^2 + 6x \)[/tex] factors completely to [tex]\( 3x(7x + 2) \)[/tex].
