Sure, let's solve the given equation step-by-step to find its equivalent form:
1. Start with the given equation:
[tex]\[
x^2 - 6x = 8
\][/tex]
2. We want to complete the square. To do this, we first move the constant term to the other side:
[tex]\[
x^2 - 6x - 8 = 0
\][/tex]
3. Next, isolate the quadratic and linear terms:
[tex]\[
x^2 - 6x = 8
\][/tex]
4. To complete the square, take half the coefficient of [tex]\( x \)[/tex], square it, and add it to both sides of the equation. Here, the coefficient of [tex]\( x \)[/tex] is [tex]\( -6 \)[/tex], half of it is [tex]\( -3 \)[/tex], and squaring it gives [tex]\( 9 \)[/tex]:
[tex]\[
x^2 - 6x + 9 = 8 + 9
\][/tex]
5. Now simplify the equation:
[tex]\[
x^2 - 6x + 9 = 17
\][/tex]
6. The left side of the equation is now a perfect square:
[tex]\[
(x - 3)^2 = 17
\][/tex]
7. The equation [tex]\((x - 3)^2 = 17\)[/tex] is a completed square form and is equivalent to the original equation.
Therefore, among the given options, the equivalent equation is:
[tex]\[
\boxed{D. (x - 3)^2 = 17}
\][/tex]
