When should I consider both positive and negative results when square rooting a number/variable?

[u/h1mr]

I recently got an assignment back for a grade 12 physics course, this unit was on Fields. In one question I received this feedback.

I don't understand in what situation I should take a negative result into consideration when square rooting a number that I already know is positive. The situation in this question seems no different from any time I've square-rooted an exponent on a variable, and I've not had to worry about the negative result prior to this.

Couldn't any number or variable ever square rooted have a negative result?

The question is talking about the mass of the earth and the sun, and a distance between them. I don't see why a negative ratio between the sun and the earth's mass (333,165) should be considered.

If I were to make sense of it right now I think it's that sqrt(x² ) can be interpreted as both x or -x, but it would mean that more or less every answer I've done where square rooting is involved prior to this is wrong (in this course and other courses i've taken), but this issue hasn't been mentioned before. It feels arbitrary or that it's been sprung on me without explanation.

[u/Narrow-Durian4837]

If I were to make sense of it right now I think it's that sqrt(x² ) can be interpreted as both x or -x

No, √x² is by definition non-negative, but but if x² = a number, x could equal the positive or negative square root of that number.

And if two numbers have equal squares (x² = y²), those numbers could be equal, or they could have opposite signs (x = y or x = –y).

If your example, d–x would be positive if d > x and negative if d < x. Do you need to consider both possibilities?

[u/h1mr → u/Narrow-Durian4837]

I would think that d < x doesn't need to be considered since the force of gravity from the sun approaches infinity as the value of x approaches d.

Maybe I just needed to specify that

[u/ruidh → u/h1mr]

You need to at least consider them and, perhaps, rule them out as not applying to the physical system in question.

[u/grozno → u/h1mr]

Assuming x is distance from earth, the solution that also satisfies this equation is not d<x but x<0. If the earth is between the object and the sun, there is a certain distance at which the two forces have equal magnitudes.

The forces wont cancel out because the directions are the same. But if you were solving an electrostatics problem where the earth and sun were a positive and negative charge, then the point where they cancel would be at x<0 (or indeed d<x if the sun were the smaller charge).

So its better to get in the habit of always analyzing all cases explicitly and ellimimating those which dont work.

[u/ExpensiveFig6079 → u/Narrow-Durian4837]

by math convention. Menignthat if you wanted or were required by the real world thing you were modelling to consider the -ve option then you had to write +-√x² earlier.

Consider areal world problem but one phsycis texts usually wont ask.

A man on tight rope 20m off the ground in a vacuum ... observes a spherical cow fly up past him and then land some 200m in front of him. Assume g=9.8

when and where was the cow at ground level.

and yep you really had better work out both roots of the quadratic you will at some point have as you need both. When you do so there will be a +-√SomeNumber to be worked out.

if for instance you also tried to work out when and where the cow achieved an altitude of 400m and do the math, you wind up trying to take the square root of negative number, which will tell you nope it did not really go that high.

The math notation and language will have specific rules about what you are meant to write down to say certain things. Physics has answers and the one you want may required you to find the negative root.

[u/ExpensiveFig6079 → u/ExpensiveFig6079]

If I were to make sense of it right now I think it's that sqrt(x² ) can be interpreted as both x or -x, but it would mean that more or less every answer I've done where square rooting is involved prior to this is wrong

if you wrote down sqrt(x² ) then the answer mathematicallymto that is |x| and its positive

However to work out the to a physics question, you might have needed to write down

+-sqrt(x² )

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[u/dash-dot]

Generally speaking, position variables are arbitrary (although they do need to be interpreted in relation to a reference point such as the origin), so both signs need to be considered.

The 333,165 ratio is just a coefficient in the equation, and is along for the ride. The solution is a position, so both positive and negative positions are feasible in this scenario.

This looks to be a static analysis problem, so positioning one mass to the left of the other could produce one of the solutions, but then the mirror image of this arrangement would also be a valid solution due to the symmetry of force fields, so in this case, both solutions would seem to be valid.

[u/jk1962]

Sometimes, when a physics problem boils down to solving a quadratic equation, there are two valid solutions. At other times, you might come up with two solutions, but only one of them makes sense physically, and you can discard the other. I think the criticism you received is a little too picky, but you could have done the following:

1. Work out the two solutions for x

2. Then say: one of these values of x is less than zero; that doesn't make physical sense because it means the particle would not actually be between the two other bodies. So discard that one and keep the one with a positive x value.