# Why do we use plus/minus signs in front of radicals?

• tahayassen
In summary, the use of plus/minus signs in mathematics is not always used in front of radicals. The n-th root function can only produce one output, but in cases where n is an even number, the square root can be positive or negative and still give the same result. In solving equations, both cases are usually needed. Additionally, the notation of \pm and \mp is used to represent both cases separately. However, it should be noted that |x| is not the same as \pm x, so \pm|x| is not the same as \pm\pm x.
tahayassen
$$±\sqrt { { x }^{ 2 } } \\ =±|x|\\ =±±x$$

Wouldn't a plus sign or a minus sign be sufficient?

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hi tahayassen!
tahayassen said:
Wouldn't a plus sign or a minus sign be sufficient?

you mean, wouldn't one plus-or-minus sign be sufficient?

yes, you can just write "√(x2) = ±x"

The last step is either false or misleading nobody would write that. It means in one case ++x and in the other case --x which is both the same as x and not what you wrote on the line above. Plus-minus signs are used if one wants to say that there are two equations that apply in different cases, and that are only distinguished by different plus or minus signs. It's a way to express your thoughts and not a rigorous 100% correct way to write mathematics. If one wants to be clear one has to write cases like

"c=a+b if condiction X and c=a-b if condition Y."

plus/minus signs

You should also note that plus/minus signs written as in your thread title means something different.

+/- means plus or minus, but not both, ie only one will satisfy the statement.

So an object that slides up and down a vertical pole can be moving up (y +ve) or down (y -ve) but not both.

± means both plus and minus (satisfy the statement)

The equation x2 = 4 is satisfied by both x=+2 and x=-2

Plus/minus signs are not aways used in front of radicals. The n-th root is a function which means it can produce only one output. In the case where n=even number the square root is positive number which rised to the power of n gives the original number. However negative numbers risen to even power give the same answer as the same positive numbers. When you solve equations you usualy need both cases.

And yes +/- sign mean both cases separately for example :

$(x \pm y)^2=x^2 \pm 2xy+ y^2$ This is basicaly the two cases in one equation :
$(x + y)^2=x^2 + 2xy+ y^2$
$(x - y)^2=x^2 - 2xy+ y^2$

tahayassen said:
$$±\sqrt { { x }^{ 2 } } \\ =±|x|\\ =±±x$$
This last step is incorrect. $\pm\sqrt{x^2}$ is the same as $\pm |x|$ but |x| is NOT the same as "$\pm x$" so $\pm |x|$ is NOT the same as "$\pm\pm x$".

Wouldn't a plus sign or a minus sign be sufficient?
How would you chose which?

## 1. Why do we use plus/minus signs in front of radicals?

The plus/minus sign in front of a radical indicates that there are two possible solutions for the equation. This is because a radical represents the square root of a number, and both the positive and negative square root of a number are valid solutions.

No, using only the positive sign in front of radicals would restrict the solutions to only positive values, and there are cases where negative values are also valid solutions. The plus/minus sign allows for both positive and negative solutions to be considered.

We use the plus/minus sign in front of radicals when solving equations that involve finding the square root of a variable. This is because the variable could have both a positive and negative value as its square root, so the plus/minus sign allows us to consider both possibilities.

## 4. Can the plus/minus sign be used with other types of radicals besides square roots?

Yes, the plus/minus sign can also be used with other types of radicals, such as cube roots or higher order roots. This is because the concept of having both a positive and negative solution still applies.

## 5. Is there a specific order for using the plus/minus sign with other mathematical operations?

No, there is no specific order for using the plus/minus sign with other mathematical operations. It is typically used after other operations, such as addition or subtraction, have been performed on the equation.

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