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Why the "±" on the one side?


In the "solving by square-rooting" section of the "solving quadratics" lesson, we had the following problem and solution:

  • Solve x2 – 4 = 0.
    • x2 – 4 = 0

      x2 = 4

      sqrt(x) = ± sqrt(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:

 

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    sqrt(x^2) = sqrt(4)

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:

    sqrt(x^2) = abs(x)

Because of this technical consideration, the equation simplifies as:

    x2 = 4

    sqrt(x^2) = sqrt(4)

    | x | = 2   Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved

But x could be positive or negative (though not zero, obviously). To solve this absolute-value equation, you have to consider both 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, the "plus-minus" 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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Cite this article as:

Stapel, Elizabeth. "Why the '±' on the one side?" Purplemath. Available from
    http://www.purplemath.com/modules/whyplus.htm. Accessed
 

 



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