Help with SAT Math Question

Test Preparation SAT Subject Test Preparation
1

In the New Sat Practice Tests:

Test #4, Section 3, Question 4:

Sqrt(x - a) = x - 4

If a = 2. what is the solution set to the equation above?
…

I narrowed down the answer choices to
A) {3, 6}
D) (6)

I squared both sides and found two possible answers: 3 and 6.

If x is 6, the equation simplifies to: ±2 = 2, which is correct.
If x is 3, the equation simplifies to ±1 = 1, which is supposedly correct?

The test says that the correct answer is D). Stating that 3 is an extraneous solution… How is it an extraneous solution?
I am solving for the square root of 1, and because this is part of an equation, there are two answers: +1 and -1.

Why is the answer D) rather than A)?

2

@musicer The square root of k denotes the nonnegative number whose square is k. So sqrt(4) = 2, although the equation x^2 = 4 has two solutions.

I’m this case, x = 3 is an extraneous solution since sqrt(1) is not -1.

3

This is true… However, I have come across SAT Questions where the answer to finding a radical is both the negative and positive.

How am I supposed to know when a radical should have a plus or minus, especially with this inconsistency on the test?

4

@musicer examples?

If it contains a square root of a nonnegative number, you take the non-negative (positive if nonzero) solution, e.g. sqrt(4) = 2, sqrt(x^2) = |x|.

Another example: the equation x^2 = 100 is not equivalent to x = 10. It is easy to “square root” both sides and make an error that way. Rather it is actually equivalent to |x| = 10.