Is the $\mp$ symbol in the quadratic formula redundant?

Question

Assume $D = d^2$.

Correct me if I am wrong.

When taking a square root, the output can be either positive or negative (i.e., $\sqrt D = \mp d$). So does that not make the second term in the Quadratic Formula, $$\frac{-b \mp \sqrt{b^2 -4ac}}{2a},$$ redundant? The $\sqrt{b^2 - 4ac}$ is already either positive or negative, so the $\mp$ is implied, right?

I have seen many articles arguing both that a square root returns absolute value and that a square root returns either a positive or a negative. A square root can return either positive or negative, so would it not be impossible for the output to be an absolute value because the negative output violates the definition of an absolute value?

The square root is a function, so it can return only one value, which must be nonnegative. The articles must be referring to the solutions to $x^2=a$, in which the solutions are both $\sqrt a$ and $-\sqrt a$. — bobeyt6, Oct 12 '22 at 23:12
Does this answer your question? Why do I get one extra wrong solution when solving $2-x=-\sqrt{x}$? It's the same principle. — bobeyt6, Oct 12 '22 at 23:15

score 2 · Accepted Answer · answered Oct 12 '22 at 23:21

It's a bit confusing, but there are "the square roots of a number", and "the square root function", and they are slightly different things.

As an example, $4$ has two square roots, $+2$ and $-2$, but the square root function, which is what $\sqrt{\cdot}$ refers to, gives a value of $2$ when evaluated at $4$. It always returns the positive square root, and could be called "the positive square root function", except that's too many words to be saying all the time.

More accurately, the principal square root function, which may be a complex number. — Dan, Oct 12 '22 at 23:42

Is the $\mp$ symbol in the quadratic formula redundant?

1 Answers1