When is the root of a number both negative and positive?

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Homework Help Overview

The discussion revolves around the concept of square roots and their properties, particularly focusing on when a root can be considered both negative and positive. Participants are exploring the distinction between the principal square root and the general solutions of equations involving squares.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the nature of square roots, particularly the principal square root versus the general solutions of equations. There is confusion about why the square root of a number is defined as nonnegative and how this relates to functions that can yield both positive and negative outputs.

Discussion Status

The discussion is active, with participants clarifying the difference between the principal square root and the general solutions to equations. Some guidance has been provided regarding the notation and interpretation of square roots, though multiple interpretations of the concepts are still being explored.

Contextual Notes

There is mention of a test question that has led to confusion regarding the expected answers, highlighting the importance of understanding the definitions and properties of square roots in different contexts.

BruceSpringste
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Homework Statement



I have a simple problem with roots and absolute values. When is the root of a number both negative and positive? Is only the equation of a number say f(x) = √x both the negative root and the positive root?

Homework Equations



If a = 1; b = -2, och x = a2√(ab-b2+2)

Why is x only 2 and not -2 aswell?

However if it were a function say f(x) the answer would be 2 and -2 right?

Edit: For clarification how come √4 = 2 but f(4) = √4 = 2 and -2
 
Last edited:
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BruceSpringste said:

Homework Statement



I have a simple problem with roots and absolute values. When is the root of a number both negative and positive?
Given that we're talking about real roots of real numbers, an even root is not both negative and positive.
BruceSpringste said:
Is only the equation of a number say f(x) = √x both the negative root and the positive root?
The symbol √x represents the principal square root, which is a nonnegative number. For example, √4 = 2.
BruceSpringste said:

Homework Equations



If a = 1; b = -2, och x = a2√(ab-b2+2)

Why is x only 2 and not -2 aswell?
I think you have a typo. If a = 1 and b = -2, then the quantity inside the radical is 1*(-2) - (-2)2 + 2 = -2 - 4 + 2 = -4.
BruceSpringste said:
However if it were a function say f(x) the answer would be 2 and -2 right?
No. A function can have only one output value. Otherwise it's not a function.
 
Yes there was a typo! x = a2√(ab+b2+2)

When in a test do you know the difference between the principal square root and the root which gives you the answers +2 and -2? Because the answer from the test which the question was taken is 2. Why not both 2 and -2? Also I understand functions only have one y value but it can have to x values. It can't have two answers but it can have to inputs right?
 
Last edited:
It is almost always the principal square root you should be concerned with.

You might have confusion between √x2 and √4 .

√4 = 2 but √x2 = |x|

So if you have x2 = 4 ,you must have seen the solution as x= +2 ,-2 .

This is because when you take square root on both the sides ,on the left you get |x| and on the right you get 2.

x2 = 4
√x2 = √4
|x| = 2
x=+2,-2
 
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@Tanya Sharma thank you very much, your explanations are spot on!
 

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