Pwn the sat 4th edition page 36 question 3

I am tutoring my sophomore on her SAT math and we are going through your book a page at a time.

This problem is under back solving.

Given the amount of time you need to spend multiplying the left hand side of the equation. Wouldn’t it be faster to solve for a than plugging in the values?How would you approach this differently?

Thanks

The question:

I get the “Wouldn’t it be faster…” question a lot about both the plug in and backsolve techniques, and my stock answer is maybe. In this case, I agree with you that FOILing the left hand side until you get the right answer (as many as 3 times—if you fail 3 times then you know the 4th choice is correct) might take a minute. I also agree that if you know what you’re doing with complex numbers, solving for *a* is fairly straightforward. To make a stronger case for backsolving here, I maybe should have made the question a tiny bit tougher. Like this, perhaps:

As I say at the top of that drill, though, I think it’s worth practicing the techniques, *even if they feel slower or actually are slower*, because if you really internalize the technique, it can be an escape hatch for you on test day when you encounter a question you can’t figure out the math way. If you never practice backsolving because it always feels too slow, then it probably won’t be there for you in a pinch when you need it most.

Here’s how backsolving would look for me on this question. I’d start with C, since I pretty much always start with C:

That’s obviously false, but it’s CLOSE! Being 1 off would almost certainly clue me in to trying choice B next, and I’d follow the pattern of the FOILing I just did, so I’d be a little faster the 2nd time.