Question

determine which answer choice below is equivalent to the given expression by completing the square.\\(x^2 - 6x - 6\\)\\(\text{a. } (x - 3)^2 - 24\\)\\(\text{b. } (x - 3)^2 - 15\\)\\(\text{c. } (x - 3)^2 - 9\\)\\(\text{d. } (x - 3)^2 - 6\\)

Explanation:

Step1: Isolate x terms

$x^2 - 6x - 6 = (x^2 - 6x) - 6$

Step2: Complete the square for $x^2-6x$

Take half of -6: $\frac{-6}{2}=-3$, square it: $(-3)^2=9$. Add and subtract 9 inside the parentheses.
$(x^2 - 6x + 9 - 9) - 6 = (x^2 - 6x + 9) - 9 - 6$

Step3: Rewrite as perfect square

$x^2 - 6x + 9 = (x-3)^2$, so simplify constants:
$(x-3)^2 - 15$

Answer:

B. $(x - 3)^2 - 15$