Question

which value of a would make the following expression completely factored?
$x^2 - a$
12
36
49
81

Explanation:

Step1: Recall difference of squares rule

A binomial $x^2 - b$ can be factored as $(x-\sqrt{b})(x+\sqrt{b})$ if $b$ is a perfect square. For the expression to be completely unfactorable (already in fully factored form over integers), $a$ must NOT be a perfect square.

Step2: Identify perfect squares

Check each option:

• $36 = 6^2$, $49 = 7^2$, $81 = 9^2$ (all perfect squares)
• $12$ is not a perfect square ($\sqrt{12}=2\sqrt{3}$, irrational)

Step3: Verify factoring status

If $a=12$, $x^2-12$ cannot be factored into integer binomials, so it is completely factored. For the other options, the expression would factor further.

Answer:

12