Let's break down each problem step by step to understand the factorization provided:
### Problem 1
Given the expression:
[tex]\[ x^2 - 6x + 9 \][/tex]
To factor this, notice the coefficients follow a perfect square trinomial pattern:
[tex]\[ a^2 - 2ab + b^2 = (a - b)^2 \][/tex]
Here,
[tex]\[ a = x \][/tex]
[tex]\[ b = 3 \][/tex]
[tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
### Problem 2
Given the expression:
[tex]\[ 4x^2 - 4x + 1 \][/tex]
We can identify another perfect square trinomial:
[tex]\[ (2x)^2 - 2(2x)(1/2) + (1/2)^2 = (2x - 1/2)^2 \][/tex]
Simplifying:
[tex]\[ 4x^2 - 4x + 1 = (2x - 1)^2 \][/tex]
### Problem 3
Given the expression:
[tex]\[ 39x^2 + 12x + 4 \][/tex]
Upon analysis, this expression cannot be factored into a perfect square trinomial that fits the given problem's form. Therefore, we leave it in its original quadratic form.
### Problem 4
Given the expression:
[tex]\[ 4x^2 + 16xy + 16y^2 \][/tex]
We see a perfect square trinomial with:
[tex]\[ (2x + 4y)^2 \][/tex]
Specifically,
[tex]\[ a = 2x \][/tex]
[tex]\[ b = 4y \][/tex]
[tex]\[ 4x^2 + 16xy + 16y^2 = (2x + 4y)^2 \][/tex]
### Problem 5
Given the expression:
[tex]\[ 16a^2 - 24ab + b^2 \][/tex]
This fits a perfect square trinomial pattern:
[tex]\[ a = 4a \][/tex]
[tex]\[ b = 3b \][/tex]
[tex]\[ 16a^2 - 24ab + b^2 = (4a - 3b)^2 \][/tex]
So, our factorizations for each problem are:
1. [tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
2. [tex]\[ 4x^2 - 4x + 1 = (2x - 1)^2 \][/tex]
3. [tex]\[ 39x^2 + 12x + 4 \][/tex] (cannot be factored into a simplified squared form)
4. [tex]\[ 4x^2 + 16xy + 16y^2 = (2x + 4y)^2 \][/tex]
5. [tex]\[ 16a^2 - 24ab + b^2 = (4a - 3b)^2 \][/tex]
I hope this helps you understand the step-by-step factorization process for these problems!
