The set of all real numbers $x$ such that $\sqrt{x^2}=-x$

Question

The questions goes as

The set of all real numbers $x$ such that $\sqrt{x^2} = -x$ consists of

a. Zero only
b. Nonpositive real numbers only
c. positive real numbers only
d. all real numbers
e. no real numbers

I chose b but I don't completely understand. I learned that the term "nonpositive real numbers" means that $0$ is included with all the numbers to negative infinity. I know that all positive real numbers have two square roots. Can someone explain this to me please?

Answer 1

You have $\sqrt{x^2} = |x|$. Now the answer becomes clear.

Answer 2

The answer is $(b)$.

Since $\sqrt x^2=|x|$ and the definition of $|x|$ when $x<0$ is $-x$, which is usually how its defined in a Calculus I course.

Answer 3

For the purposes of the SAT, I encourage a number substitution approach, remembering to choose common exceptional cases.

Choose $x = 0$, a choice suggested by choice (A): Then $\sqrt{x^2} = \sqrt{0^2} = 0$, and $-x = -0 = 0$, so the equation checks out. This eliminates (C) and (E).

Next, choose $x = 1$: we get $\sqrt{x^2} = \sqrt{1^2} = 1$, but $-x = -1$ which is false, thus eliminating (D).

Finally, choose $x = -1$: we get $\sqrt{(-1)^2} = \sqrt{1} = 1$, and $-x = -(-1) = 1$, which works, eliminating choice (A). Therefore, the answer must be choice (B).

It is a longer approach than knowing that $\sqrt{x^2} = |x|$, but on the SAT, sometimes we don't always know the higher-level principle to be able to answer questions immediately, and instead the student can rely on lower-level principles as in this case, to guide oneself to a correct answer even if a proof is not known.

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In regard to the way square roots are used on the SAT, it is made clear in the instructions for the mathematics sections that the notation $\sqrt{x}$ is intended to denote the unique nonnegative number whose square equals $x$; i.e., the nonnegative square root of $x$. The SAT I does not require knowledge of complex numbers.