To factor the quadratic expression [tex]\( ax^2 + bx + c \)[/tex], follow these steps. Here, we are given the coefficients:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = 1\)[/tex]
We need to find the factors of this quadratic expression.
### Step 1: Write down the quadratic expression
The given quadratic expression is:
[tex]\[
x^2 + 2x + 1
\][/tex]
### Step 2: Find two numbers that multiply to [tex]\(ac\)[/tex] and add up to [tex]\(b\)[/tex]
In this quadratic expression, [tex]\(a = 1\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = 1\)[/tex].
First, multiply the leading coefficient [tex]\(a\)[/tex] by the constant term [tex]\(c\)[/tex]:
[tex]\[
ac = 1 \times 1 = 1
\][/tex]
Now, we need to find two numbers that multiply to [tex]\(1\)[/tex] (which is [tex]\(ac\)[/tex]) and add up to [tex]\(2\)[/tex] (which is [tex]\(b\)[/tex]).
The two numbers that satisfy these conditions are [tex]\(1\)[/tex] and [tex]\(1\)[/tex] because:
[tex]\[
1 \times 1 = 1 \quad \text{and} \quad 1 + 1 = 2
\][/tex]
### Step 3: Rewrite the middle term using the two numbers found
We can rewrite the quadratic expression by splitting the middle term [tex]\(2x\)[/tex] into [tex]\(x + x\)[/tex]:
[tex]\[
x^2 + x + x + 1
\][/tex]
### Step 4: Factor by grouping
Now, group the terms in pairs:
[tex]\[
(x^2 + x) + (x + 1)
\][/tex]
Factor out the greatest common factor from each pair:
[tex]\[
x(x + 1) + 1(x + 1)
\][/tex]
### Step 5: Factor out the common binomial factor
Notice that [tex]\((x + 1)\)[/tex] is a common factor in both groups. Factor [tex]\((x + 1)\)[/tex] out:
[tex]\[
(x + 1)(x + 1)
\][/tex]
### Step 6: Write the final factored form
The factored form of the quadratic expression is:
[tex]\[
(x + 1)^2
\][/tex]
So, the quadratic expression [tex]\(x^2 + 2x + 1\)[/tex] factors to [tex]\((x + 1)^2\)[/tex].
