Why does following $\sqrt{x^2} = \lvert x\rvert$ appear to give an extra solution?

Question

$$a=4$$ $$a^2=16$$ $$\sqrt{a^2}=\sqrt{16}$$ $$|a|=4$$ $$a=\pm4$$ Why does it appear as though $a$ can equal $-4$, at the end, even though it was only positive $4$ at the beginning?

Answer 1

A good way to debug such things is to plug $a=-4$ into each of your equations to find the point where it first became true for the unwanted value.

You will find that $a=4$ is false when $a$ is $-4$, but $a^2=16$ is true when $a$ is $-4$. So this is the step where the extraneous solution was introduced.

$\sqrt{x^2}=|x|$ does not have anything to do with it.

Answer 2

HINT: we get $$(a)^2=a^2$$ or $$(-a)^2=(-1)^2a^2=a^2$$ the same result.

Answer 3

$$(-4) \neq 4$$

$$(-4)^2 = 4^2$$

$x^2$ is not an injective function.

Answer 4

Consider this line of reasoning:

Suppose $a = 4.$ It follows that $a = 4$ or $a = -4.$

Do you see anything wrong with that line of reasoning? I hope you do not, because the reasoning is perfectly valid. It would be very strange to find out that $a=4$ and yet neither $a=4$ nor $a = -4$ was true.

Your do not explicitly state how you developed your sequence of equations, but in the absence of explicit language or logical symbols to indicate how the statements relate to each other, we usually take such a sequence of statements to mean that each statement can be justified by the ones that came before it. Also, we read $a = \pm 4$ as the statement, "$a = 4$ or $a = -4.$"

Given the initial assertion, $a=4,$ it is very simple to justify that $a = 4$ or $a = -4.$ That's what your sequence of statements did.

What you cannot do is to use the assumption that $a = 4$ or $a = -4$ to justify a statement that $a=4.$ In other words, you can't just take your sequence of statements and write them down in reverse order, starting with $a = \pm 4$ and ending with $a=4.$ That doesn't make sense. The reason it doesn't work is that $a=4$ is sufficient to show that $a^2 = 16,$ but $a^2 = 16$ is not sufficient to show that $a = 4.$