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Why the "±" on the one side?

We had the following problem and solution:

• Solve x² – 4 = 0.

x² – 4 = 0

x² = 4



x = ± 2

Then the solution is x = ± 2.

...and I explained the form of the solution by saying:

Why the "±" ("plus-or-minus") sign? Because it might have been a positive 2 or a negative 2 that was squared to get the 4.

However, if you want to get really technical, the explanation goes like this:

Suppose you are given the equation "x² = 4" and told to solve. When you square-root both sides, you get this:

That is, technically-speaking, you don't have a "±" on the square root sign on the right. However— The technical definition of "the square root of x squared" is "the absolute value of x". That is:

Because of this technical consideration, the equation simplifies as:

x² = 4



| x | = 2   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

But x could be positive or negative (though not zero, obviously). To solve this absolute-value equation, you have to consider cases. If x is positive, then you can take the absolute-value bars off without changing anything:

if x > 0, then | x | = x, so | x | = x = 2

On the other hand, if x is negative, then you have to change the sign on x when you take the absolute-value bars off, so you get:

if x < 0, then | x | = –x, so | x | = –x = 2

Solving this, you get that x = –2.

That is, while we place the "±" sign on the side with the number, it actually (technically) comes from the side with the variable, because of the absolute value on the variable. However, most students find it simplest just to remember that, whenever you square-root both sides of an equation, you have to remember to put a "±" on the side opposite the variable.

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