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I'm solving the following GRE problem: Solve $4x^2=-16x$

Method 1: I simply divide both sides by $4x$ :$$x=-4$$

Method 2: I solve by factoring:$$4x^2+16x=0$$

$$4x(x+4)=0$$ $$x=-4, x=0$$

Using method 1, I did not get $x=0$ as a solution. Is method 1 wrong? If so, why?

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You can't divide both sides by $4x$ if $x = 0$. –  Qiaochu Yuan Jun 6 '13 at 5:59
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4 Answers

Your first method is not wrong, but notice that you can only divide by $4x$ if $x\neq0$.

If $x=0$, then it is already a solution, and this is how you can add $x=0$ as a solution in the first method as well.

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Method 2 gives both correct answers. Method 1 did not yield $x = 0$ because when you divided both sides by $4x$, that was under the assumption that $x \neq 0$ because dividing by $0$ is not allowed arithmetically. If you wanted to correctly implement Method 1, you would have to note that $x = 0$ is a root (by inspection or another method) and then you could divide both sides by $4x$ to ascertain the non-zero root(s).

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In order to use method 1, you made the tacit (and unwarranted) assumption that $x\ne 0$. (Otherwise, you couldn't divide by $4x.$)

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As this is clear this is quadratic equation so there must be two roots.If you follow method 1,which is incorrect ,you can't get two roots because you cancel one x and also you've to make sure that $x\ne 0$ to divide on both side.

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