Question

select the correct answer.
what is the complete factorization of $x^2 - 6x + 9$?
a. $(x + 3)^2$
b. $(x + 1)(x - 9)$
c. $(x - 3)(x + 3)$
d. $(x - 3)^2$

Explanation:

Step1: Recognize perfect square trinomial

A perfect square trinomial has form $a^2 - 2ab + b^2=(a-b)^2$. For $x^2 -6x +9$, $a=x$, $2ab=6x$.

Step2: Solve for $b$

Substitute $a=x$ into $2ab=6x$: $2(x)b=6x$ → $b=3$. Verify $b^2=9$, which matches the constant term.

Step3: Apply the formula

Substitute $a=x$, $b=3$ into $(a-b)^2$: $(x-3)^2$.

Step4: Verify by expanding

Expand $(x-3)^2$: $(x-3)(x-3)=x^2-3x-3x+9=x^2-6x+9$, which matches the original polynomial.

Answer:

D. $(x - 3)^2$