x2 – x2 = x2 – x2 conundrum

Posted on 2016-09-25
Last Modified: 2016-10-02
Consider the following math example:
x2 – x2 = x2 – x2          *Note: the 2s on this line are all superscript
x (x-x) = (x-x)(x-x)
x = 2x
1 = 2

I remember a few months ago I read some article that gave an explanation for this conundrum (it was some link from, but now I cannot find that article. Can someone find that article? Or, at the very least, can someone else explain why this does not violate mathematics?
Question by:john8217
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Accepted Solution

Pavel Gushchin earned 250 total points
ID: 41815356
You can not divide both sides on (x-x), because it is equal to 0.
As you know, it's impossible to divide on 0.
LVL 82

Expert Comment

by:Dave Baldwin
ID: 41815369
Since both sides are obviously equal to 0, I would stop right there.  Any other conclusion is a trick.
LVL 27

Expert Comment

ID: 41815376
The above two answers are correct.
In your post you said
x (x-x) = (x-x)(x-x)
x = 2x
in any case, I do not see how you got the second line above
LVL 17
ID: 41815420
The right hand side of the second line should be (x+x)(x-x)
Just a sign change.
LVL 27

Expert Comment

ID: 41815441
good. That clears up two difficulties. The left hand side is still dividing by 0, a prohibited operation.
LVL 13

Assisted Solution

by:Norm Dickinson
Norm Dickinson earned 250 total points
ID: 41815857
This is an identity. This shows that something is equal to itself, and no matter what kind of mathematical operation you perform, if you perform it the same on both sides of the equation, both sides will still equal each other. The question itself goes poorly when the example is added. When both sides start out identical, both sides must remain identical after each operation. Identical = Identity. There is nothing to change that. 0 = 0, 1 = 1, Y = Y, you can say it any way you want, but something always equals itself.

We take the original formula of x^2 - x^2 = x^2 - x^2

The first step of breaking this down can in fact be reducing each side by dividing by x. But this ends up differently than the example shows. You will always end up with the same results on both sides, when both sides start out exactly the same, not just mathematically equal.

(x^2 - x^2) = (x^2 - x^2)
           x                    x

This brings us to
x-x = x-x

0 = 0

Or substitute 3 for x. Now you have 3^2 - 3^2 = 3^2 - 3^2
This reduces to 9 - 9 = 9 - 9, or 0=0.

There is nothing that makes this difficult other than the "sample" solution above, which is incorrect. The fallacy is that the example states x^2 - X^2 = (x-x)(x-x). But if we check that math, we find that (x-x)(x-x) actually evaluates into (x^2) + (-x^2) + (-x^2) + (x^2). This breaks down to zero again when combined. So the way it should have been broken down is the same as the other side. The real answer is x(x-x).

Once again, both sides are equal and exactly the same - and the statement remains true. Any legal mathematical operation performed on both sides of this equation will show that it remains equal. That's not to say that both sides equal zero, because under the rules you can add a hundred to each side if you want. The real answer is both sides are simply equal no matter how you slice it. This is true in mathematics - if two things are exactly identical, then they are also equal. A formula is a formula is a formula, no matter which side of the equation it is on. Enjoy! (Update: Actually this should read "expression" rather than formula...)
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Expert Comment

ID: 41816457
I have seen several of these type problems over the years.  They always include a divide by zero operation somewhere which is undefined.  To go from line 2 to line 3 you have to divide both sides by (x-x)=0.
LVL 13

Expert Comment

by:Norm Dickinson
ID: 41816474
You don't have to divide by zero - in fact, you are not allowed to do so, as it exceeds your mathematical authority - just like it does mine.

You can substitute for the variable and find all kinds of numbers that work. There is more than one way to solve a single variable equation.

If you make a chart for x that runs the range of positive integers, you will see that any value will work to solve for x - it will just come out on the other end as x = x.

Zero is the only number that this equation does not solve for x, as that would result in an undefined operation.

You can also just add x to both sides, and end up with x = x.
LVL 17
ID: 41816825
>You can also just add x to both sides, and end up with x = x.
So from the question:
2+× = 1+x ?

The question was answered fully and accurately in the first comment, anything else is just adding confusion
Why post in an answered question unless you are adding something of benefit?
What you added may be true, but does not answer the question
LVL 13

Expert Comment

by:Norm Dickinson
ID: 41817112
Sorry for your confusion, I was replying to another expert, not the original question.
LVL 20

Expert Comment

ID: 41818633
I've seen a similar example that gets past the initial "0=0" problem by using a and b instead of just x.  The net result is effectively the same.

OceanExpert has the correct answer.  This is an excellent illustration (I've used a similar one many times) on how it can be essential to use the actual math steps rather than a shortcut.

The usual answer for how one gets from line 2 to line 3 (ignoring the error in the OP) is that one "cancels out the (x-x)".  "Cancelling" is a shortcut in math, not a proper step.  If you do the actual math (dividing both sides by x-x) it may be more apparent that you are dividing by 0.

Math shortcuts are useful and can work, but one needs to know where they are not allowed.

Author Closing Comment

ID: 41825748
Thanks guys

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