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$4x^2 = 4x$

If we divide both sides by $4x$ we get $x = 1$
but if we do this:

$4x^2 - 4x = 0$
$4x(x-1) = 0$

we get $x = 1, x = 0$
I guess the latter solution is the correct one, but I want to understand why the first one is wrong, and since it's wrong should I avoid using it all the time?
Thanks in advance.

Guest
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1 Answers1

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The first one comes as wrong, since you can only divide by a number if that number is not equal to zero. When trying to find all real numbers x such that $4x^2 = 4x$, 4x could equal 0. So, you can't divide by 4x.

You shouldn't necessarily avoiding using the first method all the time. It comes as fine to use if what you divide by cannot equal zero. For instance, if the problem said: "Find all x such that $4x^2 = 4x$ where 4x does not equal 0," then you could divide by 4x.

But, if the variable or equation you want to divide by could equal 0, then don't divide. Try another method.

Doug Spoonwood
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    I agree that one should in general avoid dividing and rather factories just for safety. However, I have several times broken this, but I will need to add an equation. Example: sinx = cosx. You can divide LHS and RHS by cosx to achieve tanx=1 BUT you will need to add the equation cosx=0 to the solution. – EricAm Apr 12 '13 at 14:19
  • Actually I was going to ask a similar question too. In case we divide by $\cos(x)$ shouldn't we make sure that x doesn't equal 90 or 270? – Guest Apr 12 '13 at 14:48
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    Alternatively, you can distinguish cases: (a) Assume $x\ne0$, divide by $4x$ and conclude $x=1$. (b) Assume $x=0$ and check manually whether it solves the original equation (here, it does). In summary, you find the solution $x=1$ from (a) and $x=0$ from (b). – Hagen von Eitzen Apr 12 '13 at 15:53
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    @AdamYac It's NOT just for safety. If you divide by 0, you might still end up with a true conclusion, but the reasoning you've used is invalid. With your example you can just say that if cos x does not equal 0, then you can divide both sides by cos x and get tan x=1. If cos x=0, then x belongs {90, 270, 450, ..., (90+180x)} in degrees. Tan x does not exist at those values. – Doug Spoonwood Apr 12 '13 at 16:40