why sqrt(36) is just positive 6

When we want to evaluate the square root of a number, say 36, the answer is just positive 6. It is a common math mistake for students to say the answer is both positive 6 and negative 6. It is both positive and negative 6 when we want to solve the quadratic equation x^=36

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#math #algebra #mathbasics

[bprpmathbasics]

sqrt(20)-sqrt(5) = ?
(A) sqrt(15) (B) sqrt(10) (C) sqrt(5) (D) sqrt(2.5)

[polarchristin]

In Russian math we have arithmetic root, which is always non-negative, and algebraic root, which has no such limitation

[ChavoMysterio]

x^2=36 is an absolute value in nature because there are two cases in this, thus |x|=6.

[sxkjknjw]

To settle the debate, my high school teachers used to tell us that _by definition_, the square root is the positive term that multiplied by itself give the radicand

[beardydave]

My immediate thought was "what about the quadratic function which has a plus-minus before the sqrt?" But as you rightly say, we are not solving a sqrt, we are trying to solve an equation.

[zevfarkas]

There is a convention that (radical sign)(x) gives the positive root. The convention could instead have been the negative one (similar to the way we ended up with electrons being negatively charged). It's even more fun when you start dealing with stuff like cube roots, or when the thing under the radical sign is negative.

[kingbeauregard]

If you have to solve "x^2 = 36", turn it into "x^2 - 36 = 0" and factor it to "(x - 6)(x + 6) = 0". Both answers emerge automatically. In summary, to find all the roots, factor.

That especially holds with something like "x^3 = 36x", where a person might be tempted to cancel out the x. Nahhh, factor it: "x^3 - 36x = 0" which becomes "x(x-6)(x+6) = 0". There, now you get all three roots.

[charlesmoore]

I’m too old (82) to undertake such a project, BUT . . .

IF it were possible, I would love to study under him. If I attended regularly, maybe I’d discover what he does with the ball.

I’m serious! This man is an excellent instructor. Patient. Methodical. Even his heavily accented English is delivered clearly. I hope his students pay attention and appreciate the opportunity they have!

[pikabuu]

A lot of people make that mistake idk why. It's true that every square has a positive and a negative square root, however the symbol √ denotes the positive square root. That's a convention. No symbol exists for the negative square root so we must use -√ if we're calculating the negative square root of a number. When we calculate √ it's always the positive square root!

For example: √16=4 and -√16=-4
If √16 = 4 or -4 and -√16 = 4 or -4 that wouldnt make a lot of sense.

Life would be easier if mathematicians would introduce a new symbol for the negative square root so people don't mistake √ = positive square root for both the positive and negative square root.

when we solve for x, f.e. x^2 = 4 <=> x = +√4 or -√4 we get 2 solutions.
Notice that for the negative square root we write -√4, the -√ equals the negative square root and +√ or √ the positive square root.

√ = positive square root, -√ = negative square root.
So when we calculate √36, we calculate the positive square root = 6

[colonelmoustache]

Thats why when trying to evaluate with complex solution it may be ambiguous because there is not really any "positive" and "negative"

[BulentBasaran]

The reason why we want only one value for sqrt is because we want to simplify things and we want sqrt to be a function. Functions by definition can have only one value for each input.

[PrashanthS-fnme]

Simple, short, awesome explanation, very easily understood..This is how we want the explanation to be...Cheers!

[atticmusicgggg]

It's not really about "why", math is analytic proposition, we just defined it to be that way, so right/wrong is not particularly useful in this situation. Whether to include +/- depends on the application whether including the "-" case is *useful*.

[calingligore]

X is not just +/-6. It’s +/-sqrt(36). This further goes to show that we really mean it when we say that the definition of the square root is that it gives only the positive root. And yes we the humans came up with this arbitrary definition and we could have very well made it so it gave us only the negative root

[yoka]

sqrt(x^2) cancels out to |x|, so |x| = 6 has two roots
Even more sane approach would be to ALWAYS FACTOR
In this example x^2 - 36 = 0 can be factored into (x+6)(x-6)=0, giving us two solutions

[hypnometal]

I have never heard it explained like this. Thank you!

[Steve_Stowers]

I just now wondered whether some of the confusion is because some languages (like English) have definite and indefinite articles but others don't. In English, I can say "THE square root" when I mean the principal square root, or "A square root" when I mean either of the two square roots of a number (positive or negative).

[mensaswede]

For some reason, this explanation feels less than satisfying.

[Misteribel]

I believe the term for this is the “principal square root” in English, right? The √ symbol is specifically described as an operation to return the principal square root.

[toydotgame]

That explains why _y = x^2_ and _y = sqrt(x)_ don't both mirror over the axis (only x^2 does)!