To factor the quadratic expression [tex]\( x^2 - 2x + 1 \)[/tex] into the product of two binomials, follow these steps:
1. Identify the quadratic expression:
[tex]\[
x^2 - 2x + 1
\][/tex]
2. Recall the standard form of a quadratic expression:
[tex]\[
ax^2 + bx + c
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 1 \)[/tex].
3. Find two numbers that multiply to [tex]\( ac \)[/tex] (the product of the coefficient of [tex]\( x^2 \)[/tex] and the constant term) and add up to [tex]\( b \)[/tex] (the coefficient of [tex]\( x \)[/tex]):
- Here, [tex]\( a \times c = 1 \times 1 = 1 \)[/tex].
- We need two numbers that multiply to [tex]\( 1 \)[/tex] and add up to [tex]\( -2 \)[/tex].
4. Determine the numbers:
- The numbers are [tex]\( -1 \)[/tex] and [tex]\( -1 \)[/tex] because:
[tex]\[
(-1) \times (-1) = 1 \quad \text{and} \quad (-1) + (-1) = -2
\][/tex]
5. Write the quadratic in terms of these numbers:
[tex]\[
x^2 - 2x + 1 = (x - 1)(x - 1)
\][/tex]
6. Verify by expansion (using FOIL method: First, Outer, Inner, Last):
[tex]\[
(x - 1)(x - 1) = x^2 - x - x + 1 = x^2 - 2x + 1
\][/tex]
So, the factored form of the quadratic expression [tex]\( x^2 - 2x + 1 \)[/tex] is:
[tex]\[
( x - 1 )( x - 1 )
\][/tex]
Therefore, the answer is:
[tex]\[
(x - 1)^2
\][/tex]
