Certainly! Let's factor the trinomial step-by-step.
We are given the trinomial:
[tex]\[ 36x^2 - 72x + 36 \][/tex]
We are asked to factor it in the form:
[tex]\[ ([?]x - \square)^2 \][/tex]
Here are the steps to factor the trinomial:
### Step 1: Identify and Factor Out the Greatest Common Factor (GCF)
First, let's notice that each term in the trinomial has a common factor. In this case, the GCF is 36.
### Step 2: Factor Out the GCF
[tex]\[ 36x^2 - 72x + 36 = 36(x^2 - 2x + 1) \][/tex]
### Step 3: Factor the Quadratic Expression
Next, let's focus on the quadratic expression inside the parentheses:
[tex]\[ x^2 - 2x + 1 \][/tex]
### Step 4: Recognize a Perfect Square Trinomial
Notice that [tex]\( x^2 - 2x + 1 \)[/tex] is a perfect square trinomial. We know it is a perfect square trinomial because it fits the form [tex]\( (a-b)^2 = a^2 - 2ab + b^2 \)[/tex]. Here, in [tex]\( x^2 - 2x + 1 \)[/tex], we can see:
[tex]\[ x^2 - 2x + 1 = (x - 1)^2 \][/tex]
### Step 5: Substitute Back
Substitute back the factored form of the quadratic:
[tex]\[ 36(x^2 - 2x + 1) = 36(x - 1)^2 \][/tex]
### Final Factored Form
Thus, the fully factored form of the trinomial [tex]\( 36x^2 - 72x + 36 \)[/tex] is:
[tex]\[ 36(x - 1)^2 \][/tex]
So in the form [tex]\(( [?]x - \square)^2 \)[/tex], we have:
[tex]\[
([6]x - [6*1])^2 = (6(x-1))^2
\][/tex]
Hence, the trinomial is factored as:
[tex]\[ 36(x - 1)^2 \][/tex]
That's the factored form of the given trinomial!
