Sure, let's factor the given perfect square trinomial, [tex]\( x^2 + 6x + 9 \)[/tex].
A perfect square trinomial is a quadratic expression of the form [tex]\( a^2 + 2ab + b^2 \)[/tex], which can be factored as [tex]\((a + b)^2\)[/tex].
1. Identify the form of the trinomial:
- [tex]\( a^2 \)[/tex] corresponds to [tex]\( x^2 \)[/tex], so [tex]\( a = x \)[/tex].
- [tex]\( b^2 \)[/tex] corresponds to [tex]\( 9 \)[/tex], so [tex]\( b = 3 \)[/tex] because [tex]\( 3^2 = 9 \)[/tex].
- The middle term [tex]\( 2ab \)[/tex] should be [tex]\( 2 \times x \times 3 = 6x \)[/tex], which matches the given trinomial.
2. Confirm that it fits the perfect square trinomial form:
- Compare to the form [tex]\( a^2 + 2ab + b^2 \)[/tex]:
- [tex]\( a^2 = x^2 \)[/tex]
- [tex]\( 2ab = 6x \)[/tex]
- [tex]\( b^2 = 9 \)[/tex]
3. Factor the trinomial:
- Based on the structure [tex]\( (a + b)^2 \)[/tex]:
- Here, [tex]\( a = x \)[/tex] and [tex]\( b = 3 \)[/tex], so the trinomial [tex]\( x^2 + 6x + 9 \)[/tex] factors as [tex]\( (x + 3)^2 \)[/tex].
Therefore, the factored form of the trinomial [tex]\( x^2 + 6x + 9 \)[/tex] is:
[tex]\[ (x + 3)^2 \][/tex]
