Sure! Let's factor each perfect square trinomial step-by-step.
### 13. [tex]\( x^2 + 6x + 9 \)[/tex]
To factorize this trinomial, we need to determine if it can be written as the square of a binomial.
First, we recognize if it follows the form [tex]\( a^2 + 2ab + b^2 \)[/tex], which factors to [tex]\( (a + b)^2 \)[/tex]:
Given:
[tex]\[ x^2 + 6x + 9 \][/tex]
1. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a^2 = x^2 \Rightarrow a = x\)[/tex]
- [tex]\(b^2 = 9 \Rightarrow b = 3\)[/tex]
2. Check the middle term:
- The middle term [tex]\(6x\)[/tex] should be [tex]\(2ab = 2 \cdot x \cdot 3 = 6x \)[/tex], which matches.
Therefore, we write:
[tex]\[ x^2 + 6x + 9 = (x + 3)^2 \][/tex]
So, the factorized form is:
[tex]\[ (x + 3)^2 \][/tex]
### 16. [tex]\( x^2 - 14x + 49 \)[/tex]
To factorize this trinomial, we examine if it follows the form [tex]\( a^2 - 2ab + b^2 \)[/tex], which factors to [tex]\( (a - b)^2 \)[/tex]:
Given:
[tex]\[ x^2 - 14x + 49 \][/tex]
1. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a^2 = x^2 \Rightarrow a = x\)[/tex]
- [tex]\(b^2 = 49 \Rightarrow b = 7\)[/tex]
2. Check the middle term:
- The middle term [tex]\(-14x\)[/tex] should be [tex]\(-2ab = -2 \cdot x \cdot 7 = -14x \)[/tex], which matches.
Therefore, we write:
[tex]\[ x^2 - 14x + 49 = (x - 7)^2 \][/tex]
So, the factorized form is:
[tex]\[ (x - 7)^2 \][/tex]
### 19. [tex]\( -4x^2 - 24x - 36 \)[/tex]
This trinomial appears slightly different due to the negative sign and coefficients. We will factor out the common factor first, and then check if the remaining trinomial is a perfect square:
Given:
[tex]\[ -4x^2 - 24x - 36 \][/tex]
1. Factor out [tex]\(-4\)[/tex]:
[tex]\[ -4(x^2 + 6x + 9) \][/tex]
Now, we need to factorize the trinomial inside the parenthesis:
[tex]\[ x^2 + 6x + 9 \][/tex]
We previously recognized this form as:
[tex]\[ x^2 + 6x + 9 = (x + 3)^2 \][/tex]
So, putting it all together:
[tex]\[ -4(x^2 + 6x + 9) = -4(x + 3)^2 \][/tex]
Thus, the factorized form is:
[tex]\[ -4(x + 3)^2 \][/tex]
### Summary:
13. [tex]\( x^2 + 6x + 9 = (x + 3)^2 \)[/tex]
16. [tex]\( x^2 - 14x + 49 = (x - 7)^2 \)[/tex]
19. [tex]\( -4x^2 - 24x - 36 = -4(x + 3)^2 \)[/tex]
