A problem in logic and math?

Question

Here is the problem. What is wrong with it and why?

For any real number x:
x2 − x2 = x2 − x2

2) Factoring both sides in two different ways:
(x − x)(x + x) = x(x − x)

3) Divide both sides by x - x:
(x-x)(x+x)/(x-x) = x(x-x)/(x-x)

4) Simplified, yields:
(1)(x + x) = x(1)
(x + x) = x

5) Which is:
2x = x

6) Since this is valid for any value of x, we can plug in x = 1:
2 = 1

Answer 1

The issue lies in the fact that the original equation is 0=0.

Then, if you put in any value for x, you are dividing by 0 which as we all know, is a "no no."

Answer 2

x = 0

Answer 3

You divided by zero factor(x-x).And you know this simple fact x-x=0. Even at step 5 you have 2x-x=0 =>x=0
That is why basic properties like (additive/ commutative ) of numbers is important.
In step four, if you notice, you have placed 1 instead of 0/0 for (x-x)/(x-x).
Expression cancel out each other only when they do not present themselves as indeterminate like 0/0.
Basically numbers are just a concept and zero has a different status and aquite important one.

Answer 4

Step 3 is the problem. x - x = 0 for all x, and dividing by 0 is not defined. When you get down to step 6, "this is valid for any value of x" is not true.

Answer 5

The problem is step 3, in which you divide both sides by (x - x). By definition, x - x = 0. Dividing by zero is an illegal procedure, punishable by a five-yard penalty.

Answer 6

step 3 - x=0

Answer 7

Step 3, you divided by zero.

Answer 8

Step 3. More generally ab=ac if and only if b=c OR a=0. Here, you would get: 2x=x OR x=x: so there is no contradiction.

Answer 9

Step 3 is where the error lies, because the equation is divided by (x-x) which is zero. You can't divide anything by zero, it's undefined.

Answer 10

In step three you divide by zero (x-x). Everything after that is invalid.