When Should I Beware of Factoring Out?

[DocZaius]

Consider this equation:

$$7x*tan(4x)=5x$$

Initially, one might think to simply factor the x out of each side of the equation and be left with this:

$$tan(4x)=\frac{5}{7}$$

Well that's wrong! That is only one solution! The other solution is:

$$x=0$$

You should instead have done the following:

$$7x*tan(4x)=5x$$
$$7x*tan(4x)-5x=0$$
$$x(7tan(4x)-5)=0$$

$$x=0\ OR\ tan(4x)=\frac{5}{7}$$

Now my question is: What properties of the initial equation should give me pause and should cause me to supress my reflex to factor the x out of the equation? Although in hindsight it is obvious that plugging in "0" for x would be a solution, that doesn't explain why doing a seemingly allowed operation (dividing each side by 7x) would withhold a solution from you.

 

[LCKurtz]

Actually, it's not the "factoring out" that creates the problem. It is the fact that you divided both sides of the equation by x. Factoring, as you have demonstrated, is the correct way to do the problem. When you divide both sides by x, you are implicitly assuming that x is not zero. That is what causes you to miss part of the solution. So my advice would be either always work such problems by factoring or, if you must divide by some factor, remember to check that factor for other solutions.

 

[zgozvrm]

The problem can be avoided by always first isolating the variable to one side of the equation.

 

[DocZaius]

Yes you are right LCKurtz, I did not use the term "factoring out" correctly there.

But wow, thanks! Would you say that the following sentence is correct then?

"Cancelling a variable out of an equation implicitly assumes it is not zero"

If that is true, then I have gone through many years of math making erroneous assumptions!

Edit: I tried to think of the simplest equation possible to illustrate this issue and this is the best I could come up with. Is this correct math?

$$x*f(x)=x$$
$$x(f(x)-1)=0$$
$$x=0\ OR\ f(x)=1$$

 

[Mark44]

Yes, that's the right way to do it.