question about real numbers

Question

My question is: Solve $(x-a)(x-b)(x-c)=0$ where $a,b,c$ belong to real numbers.

By observing, I found out that $x$ can be $a$ or $b$ or $c$.

It's perfectly reasonable to say that $(x-a)(x-b)(x-c) = 0$ precisely when at least one of the terms is zero, because a product of three nonzero numbers is nonzero. Is this what you're looking for? By "method to solve", do you mean something like the quadratic formula? — Ben FL, Jun 04 '12 at 14:29

score 7 · Accepted Answer · answered Jun 04 '12 at 14:33

In the real numbers, if a product $xy$ is zero (look in your textbook to find this, usually near the "factor theorem"), then one of the terms $x$ or $y$ is zero. So if $(x-a)(x-b)(x-c)=0,$ then either $x-a=0$, $x-b=0$ or $x-c=0$. Solve these and you will find $x$ is $a$, $b$ or $c$ respectively.

score 5 · Answer 2 · edited Jun 04 '12 at 16:24

5

If $$x_1*x_2*x_3=0，$$ then $$x_1=0 \quad\text{ or }\quad x_2=0 \quad\text{ or }\quad x_3=0.$$ In your case, $$x-a=0 \quad\text{ or }\quad x-b=0 \quad\text{ or }\quad x-c=0.$$ Thus $$x=a \quad\text{ or }\quad x=b \quad\text{ or }\quad x=c.$$

edited Jun 04 '12 at 16:24

Michael Hardy

1

answered Jun 04 '12 at 14:35

Pan Yan

747

1

You know, it's not illegal to put more than one equals sign on a line :P – Ben Millwood Jun 04 '12 at 15:20
I approved M Turgeon's edit to this answer. But the use of \mbox{} in the context is (IMO) an abomination, and I changed it to \text{}. See this....um....essay: http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_%28mathematics%29/Why_you_should_never_use_%5Cmbox_within_Wikipedia It is not only to Wikipedia that those remarks apply. – Michael Hardy Jun 04 '12 at 16:26

score 3 · Answer 3 · answered Jun 04 '12 at 14:37

My interpretation of your question is this: you've found that $x=a,b,c$ solve your equation, and you're curious about whether these are the only solutions to the equation. Suppose that $x$ is not one of $a,b,c$. Then all of the terms $(x-a), (x-b), (x-c)$ are not zero. If you multiply two real numbers that are not zero, then you will end up with something that is not zero. So $(x-a)(x-b)$ is not zero. Similarly, $(x-a)(x-b)(x-c)$ is not zero. This means that any $x$ that is not one of $a$, $b$, or $c$ will not solve your equation.

score 3 · Answer 4 · answered Jun 04 '12 at 16:36

Remember that the real numbers are an integral domain i.e. a commutative ring with no zero divisors. Hence, if we have a product $rs = 0$, this means that $r=0$ or $s=0$. Hence, if you have $$(x-a)(x-b)(x-c) = 0,$$ then either $(x-a)=0$ or $(x-b)=0$ or $(x-c)=0$. Adding $a$, $b$ and $c$ to the three equations respectively gives that $$x =a \text{ or }x =b \text{ or }x =c$$

score 2 · Answer 5 · answered Jun 04 '12 at 16:32

2

As the other answers mention, your solution is correct.

The reason for this is the "Zero product property", which applies because the real numbers are a domain (or integral domain).

answered Jun 04 '12 at 16:32

The Chaz 2.0

10,464

thanks for that yes i got it.Will you please tell me how to solve:|x-2|+|x-5|=3 – mgh Jun 04 '12 at 16:36
@meg_1997 You should ask that as a separate question. – Jun 04 '12 at 16:36
You should ask that in a separate question. Or google "solving absolute value equalities". The general principle is to use the definition. – The Chaz 2.0 Jun 04 '12 at 16:36
ya i tried but it says it does not meet our quality standards – mgh Jun 04 '12 at 16:37
What @Marvis |sivraM@ said... – The Chaz 2.0 Jun 04 '12 at 16:38
@TheChaz Any reason for @Marvis|sivraM@? – Jun 04 '12 at 16:39
@Marvis: Isn't your current username a permutation of (some of) a previous username? Or am I thinking of someone else... – The Chaz 2.0 Jun 04 '12 at 16:40
Yes. It is a permutation of a subset of my name Sivaram. – Jun 04 '12 at 16:42
Hey i finally could post the question separately please someone see to it – mgh Jun 04 '12 at 17:02

question about real numbers

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