Sure, let's solve this step-by-step:
We need to find the conjugate of the given expression [tex]\(\sqrt{x-6} - 3\)[/tex].
1. Recall the Definition of Conjugates:
The conjugate of a binomial expression [tex]\(a - b\)[/tex] is [tex]\(a + b\)[/tex]. Similarly, the conjugate of an expression [tex]\(a + b\)[/tex] is [tex]\(a - b\)[/tex].
2. Identify the Terms in the Expression:
In the given expression [tex]\(\sqrt{x-6} - 3\)[/tex], we recognize that:
- [tex]\(a = \sqrt{x-6}\)[/tex]
- [tex]\(b = 3\)[/tex]
3. Form the Conjugate:
Using the definition, the conjugate of [tex]\(\sqrt{x-6} - 3\)[/tex] can be formed by changing the subtraction to addition. Thus, the conjugate is:
[tex]\[
(\sqrt{x-6}) + 3 = \sqrt{x-6} + 3
\][/tex]
4. Compare with the Choices:
Now, we need to identify which choice matches [tex]\(\sqrt{x-6} + 3\)[/tex]:
- A. [tex]\(\sqrt{x+6}+3\)[/tex] is not correct because the expression inside the square root has [tex]\(x + 6\)[/tex] instead of [tex]\(x - 6\)[/tex].
- B. [tex]\(\sqrt{x+6}-3\)[/tex] is not correct for the same reason as A.
- C. [tex]\(\sqrt{x-6}-3\)[/tex] is incorrect because it is the original expression, not the conjugate.
- D. [tex]\(\sqrt{x-6}+3\)[/tex] is the correct conjugate.
Thus, the correct choice is:
[tex]\[
\boxed{D}
\][/tex]
