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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?
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.
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).
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.
