# Characteristic Equation

$$x^2 - x - 2 = 0$$
$$(x-2)(x+1) = 0$$
$$x = -1, 2$$

in the given example, (x-2)(x+1) where x is -2 and +1 why did the last line of the example was swap?

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Don't read too much into it; it's the same as saying $x=2$ and $x=-1$. Note that if you solve $x+1=0$ for $x$, you get $x=-1$. – J. M. May 3 '11 at 13:34
$x - 2 = 0$ doesn't imply that $x = -2$. – Qiaochu Yuan May 3 '11 at 13:34
$(x-2)(x+1)=0$ $\Leftrightarrow$ $x-2=0 \lor x+1=0$ $\Leftrightarrow$ $x=2 \lor x=-1$. Is this what you're asking? (You're question does not sound very clear to me.) – Martin Sleziak May 3 '11 at 13:35
so in this case I can also choose $$x = 1, -2$$ – optimus May 3 '11 at 14:18
@liangteh, No. please make sure you understand what Martin said. – Jiangwei Xue May 3 '11 at 14:43

In the real numbers (and complex numbers, and integers, and rationals), if a product is equal to $0$, then at least one factor is equal to zero.

So for $(x-2)(x+1)$ to be equal to zero, either $x-2=0$, or $x+1=0$.

However, for $x-2$ to equal $0$ you don't need $x$ to be equal to $-2$, you need it to be equal to $2$: $x-2=0$ is equivalent to $x=2$. And for $x+1=0$ to be true, you need $x=-1$. So that's why you go from "$x$ minus $2$" to $x=2$, and from "$x$ plus $1$" to $x=-1$.

(In general, $x$ equals $a$ if and only if $x-a=0$. And $x+1 = x-(-1)$).

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@Theo: Thanks! I never proofread enough... – Arturo Magidin May 3 '11 at 16:17
Somehow I am reminded of the rant somewhere in this site (was it you?) about the use of the word "negative"... – J. M. May 3 '11 at 16:29
@J.M. I wouldn't call it a "rant", but perhaps you are thinking of the last sentence in the penultimate paragraph of math.stackexchange.com/questions/13094/… – Arturo Magidin May 3 '11 at 16:31
thank you for your clarification. – optimus May 4 '11 at 12:48