Final answer:
To find which value of 'a' makes the expression x² - a completely factored, it should be a difference of squares. The given options of 36, 49, and 81 are perfect squares, all of which would make x² - a a completely factored expression. Thus the possible correct answers are option B (36), C (49), and D (81).
Step-by-step explanation:
The question asks which value of a would make the expression x2 - a completely factored. To completely factor this expression, we want it to be a difference of squares. A difference of squares is a special factoring case where you have two perfect squares subtracted from each other, and it factors into two binomials: (x - b)(x + b), where b is the square root of a.
Looking at the options given, A. 12, B. 36, C. 49, D. 81, we want to identify which one of these values is a perfect square since a perfect square can be written as b2, which is required for a difference of squares. Among these, 12 is not a perfect square, but 36, 49, and 81 are all perfect squares, being equal to 62, 72, and 92 respectively.
If we take the expression x2 - a, and looking for perfect squares, the only factors from the given choices that would make the expression a difference of squares are 36, 49, and 81. To clarify, here is what the factoring would look like for each value of a: Each of these values for a results in a completely factored expression. Therefore, the correct answers are B, C, and D.