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For the equation: $-x^2 = -2x(3x+1)$ I can either multiply it out on the right side and get a $-6x^2-2x$ or just divide both sides by $-2x$. However, when divide out both sides, I just get one answer: $-2/5$. When I multiply it out, I get two answers: $-2/5$ and $0$.

What is wrong with dividing both sides by $-2x$??

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    $\begingroup$ There is one situation where division is forbidden. This is why you lose a solution. $\endgroup$ – Taladris Oct 12 '14 at 1:34
  • $\begingroup$ Is there a specific rule that forbids me from making these types of error that I can adhere to? $\endgroup$ – Joe Oct 12 '14 at 1:46
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    $\begingroup$ A rule, I don't know. A high sense of auto-criticism may help: before making any operation, ask yourself "Can I do this?". Auto-criticism can be substituted by a friend that reads and checks your work. I had a classmate at university that never missed a flaw or a missing gap in my writing, and was not reluctant to tell me about these. She was a real daemon, but she was really helpful $\endgroup$ – Taladris Oct 12 '14 at 1:52
  • $\begingroup$ I believe that the rule is "you cant divide by zero" $\endgroup$ – John Joy Oct 12 '14 at 15:05
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When you divide by $-2x$ you are implicitly assuming $x\neq0$ which you don't know for sure and thus remove that solution.


Note that when you divide by a non-zero value the equation still holds on each side whereas division by zero will create undefined expressions. For example, $\frac{1}{0}$ doesn't compute to any known value where there are Math SE questions on this topic if you need a reference.

Consider if the equation was $2x=x$ where if I divide by $x$, I then get $1=2$ which is a problem as that isn't true and the reason why that division isn't allowed is because the only solution is $x=0$.

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  • $\begingroup$ Then why are we allowed to divide anything by x and still get the right answer? $\endgroup$ – Joe Oct 12 '14 at 1:39
  • $\begingroup$ @Joe the point is that you aren't. You could get the wrong answer as evidenced in your post. $\endgroup$ – Cameron Williams Oct 12 '14 at 1:52
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    $\begingroup$ It’s not exactly that we’re “allowed” to do or not to do this or that in mathematics. Rather, it’s that some operations are valid, and some aren’t. Dividing by a quantity that may be zero is not valid. $\endgroup$ – Lubin Oct 12 '14 at 2:08
  • $\begingroup$ Couldn't all x's technically be zero, unless otherwise defined? Yet when we solve equations, why is it commonly taught to divide both sides by x? $\endgroup$ – Joe Oct 12 '14 at 2:23
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    $\begingroup$ @Joe: In defence to teachers who may divide by $x$, in a basic elementary level algebra class, children are unaware of even working with equations. Certain rules are bent to make it seem easier. Kids, as you know, don't like things with too many rules. But rules do exist and for good reason. So, it's always good practice to verify $x \neq 0$ before dividing with it. $\endgroup$ – Nick Oct 12 '14 at 3:32

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