Square Root Part 2

Why do we get a positive and negative answer when taking the square root of a number?

Sample:

sqrt{81} = - 9 and 9.

Why?

 

because (-9)^2=81 and (9)^2=81

 

I see. Any number squared yields a positive answer.

 

We DON'T!

In order that "square root" be a function, it must give a single value for each x. While it is true that both 4^2 and (-4)^2 are 16, the square root of 16 is 4.

The solution to the equation x^2= 7 is x= \pm sqrt(7). The reason we need to write "pm" is because sqrt(7) alone is only the positive solution.

 

Usually, the use of the symbol √ denotes the positive root, so √9=+3 (for example).
If you have an equation such as x^2=9 however, you are presumably interested in finding all possible solutions.
In that case, both the positive and the negative roots work, and both ±√9=±3.

 

When we take the square root, there is a positive and negative answer. When the square is given to us like y = sqrt{x^2}, the answer is typically a positive value.

When we come across, for example, x^2 = 16 in some calculation, we MUST take the square root rendering two answers: a positive and negative value.

You say?

 

I say we don't HAVE TO take the square root!
To solve x^2= 16, write it as x^2- 16= (x- 4)(x+ 4)= 0. Either x- 4= 0 so x= 4 or x+ 4= 0 so x= -4.

 

I said what Professor Leonard has repeatedly stated on You Tube.

 

Did he actually use, and emphasize, the word "must" in ''we MUST take the square root"? That was what I was taking exception to. There are other ways to solve such equations that are equivalent to taking the square root.

 

Your username is familiar to me. Do you also participate in other forums? I understood Professor Leonard to say the following:

1. If you come across a square root problem, for example, sqrt{9}, the answer is 3.

2. When facing a problem, say, x^2 = 9, and the student decides to take the square root on both sides, the answer is x = -3 or x = 3.

You say?

 

I understood Professor Leonard to say the following:

1. If you come across a square root problem, for example, sqrt{9}, the answer is 3.

2. When facing a problem, say, x^2 = 9, and the student decides to take the square root on both sides, the answer is x = -3 or x = 3.

You say?

 

So If you take the square root of a '9' you always get a '3' back. What you don't know is whether that '3' was originally a '-3' or a '+3'.

 

In that case, should the answer be |3|?

 

No! |3| is just 3. The solution to x^2= 9 would be written +/- 3 or as the set { 3, -3}.

 

Thank you for your input. I will search for the video where Professor Leonard explains taking a square root in more detail. Now, heading to work.

 

Thank you for your input. I will search for the video where Professor Leonard explains taking a square root in more detail. Now, heading to work.

 

Yes, the "square root of a" is DEFINED as the POSITIVE number, x, such that x^2= a. That was what I said before.

"and the student decides to take the square root of both sides" (emphasis mine). There is nothing there that says the student MUST take the square root.

 

Let me find the video where Professor Leonard explains taking the square root.