# Square Root Part 2

Discussion in 'Basic Math' started by nycmathguy, Jul 21, 2021.

1. ### nycmathguy

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

Sample:

sqrt{81} = - 9 and 9.

Why?

nycmathguy, Jul 21, 2021
2. ### MathLover1

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

MathLover1, Jul 21, 2021
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3. ### nycmathguy

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I see. Any number squared yields a positive answer.

nycmathguy, Jul 21, 2021
4. ### HallsofIvy

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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.

HallsofIvy, Nov 27, 2021
5. ### MathLover1

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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.

MathLover1, Nov 27, 2021
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6. ### nycmathguy

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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?

nycmathguy, Nov 27, 2021
7. ### HallsofIvy

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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.

HallsofIvy, Nov 28, 2021
8. ### nycmathguy

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

nycmathguy, Nov 29, 2021
9. ### Country Boy

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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.

Country Boy, Dec 15, 2021
10. ### nycmathguy

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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?

nycmathguy, Dec 15, 2021
11. ### nycmathguy

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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?

nycmathguy, Dec 15, 2021
12. ### MathLover1

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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'.

MathLover1, Dec 15, 2021
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13. ### nycmathguy

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

nycmathguy, Dec 16, 2021
14. ### Country Boy

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

Country Boy, Dec 21, 2021
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15. ### nycmathguy

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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.

nycmathguy, Dec 21, 2021
16. ### nycmathguy

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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.

nycmathguy, Dec 21, 2021
17. ### Country Boy

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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.

Country Boy, Jan 11, 2022

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