Math: a0b4103e

ID: a0b4103e

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: Watch those assumptions.

Method 1: To keep from making a speedy judgment and missing the question, observe that (1/3) will be distributed into the entire expression (x - k)(x + k), which you might recognize as a difference of squares.

(1/3)x^2 - 2 = (1/3)(x^2 - k^2)

(1/3)x^2 - 2 = (1/3)x^2 - (1/3)k^2

At this point, you can subtract out the x's and focus on just the k^2.

-2 = -(1/3)k^2

6 = k^2

√6 = k

The correct answer is (D).

This one is tricky because I could see a lot of people picking (C), √2, here, without writing anything down and thinking that they had cleverly spotted the difference of squares. You have to be careful and organize the information given.

Method 2: It might take a bit more work, but using Desmos to graph the original expression and then using a slider for k is also acceptable. 

Just slide until the graphs align perfectly. That will be the correct answer. Using the default slider, you will notice that the blue parabola is slightly above the red one at k = 2.4 and slightly below it at 2.5. Answer choices (A) and (B) are clearly incorrect. You can calculate two values, √2 and √6, to see what is close to, say, 2.45 (if you have no confidence in your knowledge of roots).

It is clear that the correct answer is (D).

Method 3: You can pretty quickly do the arithmetic to avoid assuming the wrong thing. Evaluate each answer choice with an eye on (1/3)k^2, which must equal 2.

A. (1/3)(2)^2 = (1/3)(4) = 4/3 X

B. (1/3)(6)^2 = (1/3)(36) = 12 X

C. (1/3)(√2)^2 = (1/3)(2) = 2/3 X

D. (1/3)(√6)^2 = (1/3)(6) = 2 √

The correct answer is (D).