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

[BruceSpringste]

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

 

[Mark44]

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.

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.

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.

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.

 

[BruceSpringste]

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?

 

[Tanya Sharma]

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

 

[BruceSpringste]

@Tanya Sharma thank you very much, your explanations are spot on!