Is this a mathematical fallacy or am I missing something?

[Guitarmasterx7]

I figured I might as well get some opinions on this from ye folk of the escapist being as these forums are interesting and not completely filled with idiots.

This has always bothered me, but I forgot about it until now since it's been 4 years since my last math class. So today my professor was reviewing for the final, and this was one of the problems. (I don't know how to make math symbols on the computer, so please bare with the calculator jargon)

solve:
x+(x+1)^(1/2)=5

Spoiler: so you work it out

(x+1)^(1/2)=5-x
[(x+1)^(1/2))]^2=(5-x)^2
x+1=x^2-10x+25
0=x^2-11x+24
(x-8)(x-3)=0

and eventually you get

x=8,3

Here's where it gets sketchy

at this point, you're meant to plug the answers into the problem to see if they're right. 3 checks out.

8, however turns into this

8+(8+1)^(1/2)=5
8+9^(1/2)=5
8+3=5
11=5

which is untrue, so you're supposed to discount it as an answer

BUT, clearly how the 8 came out as an answer solving the problem originally is because -3 is also a root of 9

8+(8+1)^(1/2)=5
8+9^(1/2)=5
8+(-3)=5
5=5

which is true.

When I brought this up, my professor said that it's just some mathematical rule that the square root is always associated with the sign before it, and the book solves this problem discounting the answer the same way.

I call bullshit on this, because according to that rule I could do this.
-2=-2
set x=-2
x=-2
x^2=(-2)^2
x^2=4
(x^2)^(1/2)=4^(1/2)
x=2
-2=2

I can understand the purpose of this rule for practicality's sake, but at the point where you're solving context-less numbers just for the fuck of it, you've kind of thrown practicality out the window, and it's kind of hard to swallow that an answer produced by a well working mathematical formula is wrong when there's a possible logical way the obligatory confirmation equation could have come up with the same answer.

I usually don't like to refute... well... math, out of fear of realizing that I'm a dumb-ass, but can someone confirm that I'm onto something here, or is there some huge thing I'm missing?

Edit: Alright, I get it now. Apparently, math actually does still work mathematically, and space nazis will not be invading earth with laser mounted dinosaurs, and this is all a result of misunderstanding what my professor meant by "rule." Thanks to everyone who helped clear that up.

 

[SturmDolch]

solve:
x+(x+1)^(1/2)=5

(x+1)^(1/2)=5+x

Right off the bat, these equations are not equal. It should be:
(x+1)^(1/2) = 5-x

 

[Guitarmasterx7]

solve:
x+(x+1)^(1/2)=5

(x+1)^(1/2)=5+x

Right off the bat, these equations are not equal. It should be:
(x+1)^(1/2) = 5-x

Yeah that's my bad, I wasn't doing the equation on this, I was sort of half assedly copying it from my paper, so that was a typo. Thanks for pointing that out. I changed it. The rest of the problem should be fine though (I think lol)

 

[DPutna17]

If there is one thing I know about math it's that you should take your prof's word for it. Most likely someone has brought up the same thing before and it's been dis-proved otherwise they would have found out about it by now. Especially with simple math like that were the concepts aren't exactly new. Also you made a mistake in your math the second iteration of the equation would look like this (x+1)^(1/2)=5 minus x. I don't know if that will make a difference to what your trying to point out, but it's just a heads up.

 

[careful]

your equation is equivalent to x^2-11x+24=0, using the quadratic equation you get two solutions 8 and 3, plugging these back in agree with the equation, in this form at least. so there two solutions to your equation, i dont know what your book says, but these are two solutions.

 

[Jory]

Basically, at the point where you square both sides, you're unwillingly adding extra solutions to your equation.

Thinking of this graphically might help.

(x+1)^(1/2)=5-x

Here you are finding the intersection of the graphs of (x+1)^1/2 and (5-x)

Note (x+1)^1/2 is defined only on the positives and increases at all times, 5-x is a straight line, this means there is one intersection.

Now, when you SQUARE both sides, a strange thing happens

(x+1)= x^2 - 10x + 25

Here you find the intersection of a straight line and a quadratic equation, the root you were looking for is still there, but you have also brought this negative root of the equation in to play (-3 instead of 3), so when you find both solutions, you need to check them to find the one you actually want.

 

[Guitarmasterx7]

If there is one thing I know about math it's that you should take your prof's word for it. Most likely someone has brought up the same thing before and it's been dis-proved otherwise they would have found out about it by now. Especially with simple math like that were the concepts aren't exactly new...

Yeah, I mean it makes sense that someone would have done something about it before now if there was no explanation for it, I just have never gotten one other than "it just is" which doesn't really satisfy me. Also, that + was just a paper-> computer typo. it's fixed now.

 

[careful]

Basically, at the point where you square both sides, you're unwillingly adding extra solutions to your equation.

Thinking of this graphically might help.

(x+1)^(1/2)=5-x

Here you are finding the intersection of the graphs of (x+1)^1/2 and (5-x)

Note (x+1)^1/2 is defined only on the positives and increases at all times, 5-x is a straight line, this means there is one intersection.

Now, when you SQUARE both sides, a strange thing happens

(x+1)= x^2 - 10x + 25

Here you find the intersection of a straight line and a quadratic equation, the root you were looking for is still there, but you have also brought this negative root of the equation in to play (-3 instead of 3), so when you find both solutions, you need to check them to find the one you actually want.

ahhh thanks, i was wrong.

 

[Hello My Name Is]

You also missed that the square root of 9 is not 3 but + or -3 because (-3)^2 also =9 which should make the original equation work.

8+(8+1)^(1/2)=5
8+9^(1/2)=5
8+(+or-3)=5
8-3=5 QED

 

[careful]

lolz i hate math. but i think i know why your prof said that. whenever you have an expression containing a square root, say x^1/2=5, then your supposed to interpret this as two equations:
+(x^1/2)=5
-(x^1/2)=5
since x^1/2 can either be a positive or negative number. but by purely matter of convention, by writing just x^1/2 without any further context or explanation, they mean just take the positive root, and ignore the negative root. just as a matter of convenient convention, not a axiomatic rule of mathematics.

 

[Steven True]

I think you'll be happy to know that you are not missing anything big conceptually.
When your prof says that there is a rule that the square root takes the sign before it, he is referring to a mathematical convention, not an operational or conceptual rule.

Essentially, it is a completely arbitrary convention that mathematicians have come up with to do this. Just like it is completely arbitrary to use the symbol "3" to represent the quantity three or to put the numerator over the denominator in a fraction. The mathematicians of the world got together and decided that they will use these conventions to write math down.

So when an expression says "+(x+1)^(1/2)" whoever wrote that is communicating to you that she wants you only to take the positive rote into account. It's her choice in setting up the problem.
When an expression says "-(x+1)^(1/2)" whoever wrote that is communicating to you that she wants you only to take the negative rote into account. Again, her choice.
If she wants you to take both into account she'll write +-(x+1)^(1/2) [with the "+" above the "-"].

As for your problem, the choice is up to you of what you want to communicate.
x=-2
x^2=(-2)^2
x^2=4

(x^2)^(1/2)=4^(1/2)

This says that only the positive rote needs to be taken into account.
At this point you haven't determined this. But, when you write it this way that is what you communicate to the reader. She will take that as your meaning.

It should read.

+-(x^2)^(1/2)=4^(1/2)
+-x=2
x=-2,2

Then you go back and find which roots are valid.

 

[Guitarmasterx7]

Basically, at the point where you square both sides, you're unwillingly adding extra solutions to your equation.

Thinking of this graphically might help.

(x+1)^(1/2)=5-x

Here you are finding the intersection of the graphs of (x+1)^1/2 and (5-x)

Note (x+1)^1/2 is defined only on the positives and increases at all times, 5-x is a straight line, this means there is one intersection.

Now, when you SQUARE both sides, a strange thing happens

(x+1)= x^2 - 10x + 25

Here you find the intersection of a straight line and a quadratic equation, the root you were looking for is still there, but you have also brought this negative root of the equation in to play (-3 instead of 3), so when you find both solutions, you need to check them to find the one you actually want.

Ah right, that makes a lot of sense actually. I'm assuming that it's a rule because the number that tests to be wrong is always the one involving the negative root?

 

[Jory]

Well it's not wrong either

Since in

8+(8+1)^(1/2)=5
8+9^(1/2)=5
8-3=5
5=5

Is correct if you take the negative square root, but this answer might not make sense in the real world context of a problem, or it might. Theoretically they are both correct, but when applied to a real situation one may be unrealistic.

 

[Guitarmasterx7]

snip

Oh right, so this is just a matter of semantics involved in writing equations for math classes. Ok, that's good to know. This was running me up a wall. Thanks for clarifying.

 

[Steven True]

"Ah right, that makes a lot of sense actually. I'm assuming that it's a rule because the number that tests to be wrong is always the one involving the negative root?"

Nope, not always. That is why you have to test it.
Example:

x=1
y=1/{+-[x^(1/2)]-1}

If you take the positive root you get y=1/0 , which is nonsensical.


If you take the negative root you get y=-1/2.

 

[Caspertjuhh]

do you enjoy giving fellow escapist headaches?