Question: Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified to a real number.
$\sqrt{x^2 - 12x + 36}$
Choose the correct answer below and, if necessary, fill in the answer box to complete your ch
A. $\sqrt{x^2 - 12x + 36} = \square$ (Simplify your answer.)
B. The square root cannot be simplified to a real number.
1. **State the problem:** Simplify the expression $$\sqrt{x^2 - 12x + 36}$$ and use absolute value notation if necessary.
2. **Recognize the expression inside the root:** The expression inside the square root is a quadratic trinomial $$x^2 - 12x + 36$$.
3. **Factor the quadratic:** Notice that $$x^2 - 12x + 36$$ is a perfect square trinomial because $$36 = 6^2$$ and $$-12x = -2 \times 6 \times x$$.
So, $$x^2 - 12x + 36 = (x - 6)^2$$.
4. **Rewrite the original expression:**
$$\sqrt{x^2 - 12x + 36} = \sqrt{(x - 6)^2}$$
5. **Simplify the square root of a square:**
$$\sqrt{(x - 6)^2} = |x - 6|$$
We use absolute value because the square root function always returns the non-negative value.
6. **Final answer:**
$$\boxed{|x - 6|}$$
This means the simplified form of $$\sqrt{x^2 - 12x + 36}$$ is $$|x - 6|$$.
**Summary:**
- The square root of a perfect square is the absolute value of the base.
- So, $$\sqrt{(x - 6)^2} = |x - 6|$$.
Therefore, the correct answer is option A with $$|x - 6|$$ as the simplified form.