9x^3+6x^2=0, please help!

Gew

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Hi guys!

I know this is far from rocket science, but I'm a bit retarded and my study book lacks the "Layman's terms" step-by-step guidelines for this type of equations (which usually helps me grasp), and googling the thing returns one complicated one-step-solution with lots of parenthesis and stuff after another. So please, could someone try and make me understand in really simple terms? I think I have completed the first step..

9x^3+6x^2=0 = 3x(3x^2+2x)

This is correct, is it?
Am I even supposed to start by doing this?
I can't jump straight to the "zero-product property" method, correct?
 

Gew

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Someone else suggested that starting by turning the equation into ...

x(9x^2 + 6x) = 0

.. is better practice than my fumbling ..

3x(3x^2+2x)

... but I'm really clueless here! 😢
 

topsquark

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Someone else suggested that starting by turning the equation into ...

x(9x^2 + 6x) = 0

.. is better practice than my fumbling ..

3x(3x^2+2x)

... but I'm really clueless here! 😢
You can certainly do it from there. But notice you can also factor out an extra x from what you did:
\(\displaystyle 9x^3 + 6x^2 = 0\)

\(\displaystyle (3x^2)(3x + 2) = 0\)

So either \(\displaystyle 3x^2 = 0\) or \(\displaystyle 3x + 2 = 0\)

etc.

-Dan
 

Jomo

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9x^3 + 6x^2 = 3*3*x*x*x + 3*2*x*x

What is common in 3*3*x*x*x and 3*2*x*x. The answer is obviously (since it is just a visual question) 3*x*x = 3x^2. So you factor out 3x^2 from 3*3*x*x*x + 3*2*x*x

What is common I put in bold. Write down what is in common and then between parenthesis write down what is in red.

This gives you 9x^3 + 6x^2 = 3x^2(3x+2) =0. When a product equals 0 one or more of the factors must = 0

Is there a value for x that makes 3=0?
Is there a value for x that makes x^2=0?
Is there a value for x that makes 3x+2=0?
 
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Jomo

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You can certainly do it from there. But notice you can also factor out an extra x from what you did:
\(\displaystyle 9x^3 + 6x^2 = 0\)

\(\displaystyle (3x^2)(3x + 2) = 0\)

So either \(\displaystyle 3x^2 = 0\) or \(\displaystyle 3x + 2 = 0\)

etc.

-Dan
Just out of curiosity why do so many people (qualified people) always think that (in this example) that 3 is not a factor? This is not just about you Dan. I just want to know why?
 
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Dr.Peterson

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Just out of curiosity why do so many people (qualified people) always think that (in this example) that 3 is not a factor? This is not just about you Dan. I just want to know why?
Who said that 3 is not a factor??

Not saying that 3 is a factor is very different from saying that 3 is not a factor, if that is what you are thinking.
 

Jomo

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Who said that 3 is not a factor??

Not saying that 3 is a factor is very different from saying that 3 is not a factor, if that is what you are thinking.
In my opinion when someone writes 3x^2=0 they are not thinking of 3x^2 as the product of two factors, namely 3 and x^2. If a product of factors equal 0, I feel that you should set each factor to 0 and solve. Especially when many students claim that 3=0 when x is -3
 

Otis

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… This is not just about you Dan…
I don't understand what you're talking about.

… Is there a value for x that makes 3=0?…
Why would anybody need to ask themselves whether there's any situation in which the number 3 equals 0?

:confused:
 

topsquark

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I don't understand what you're talking about.


Why would anybody need to ask themselves whether there's any situation in which the number 3 equals 0?

:confused:
Integers a modulo 3?

I have no idea why you would want to otherwise.

-Dan
 

Jomo

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Why would anybody need to ask themselves whether there's any situation in which the number 3 equals 0?

:confused:
When 3 is a factor.
 

Dr.Peterson

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In my opinion when someone writes 3x^2=0 they are not thinking of 3x^2 as the product of two factors, namely 3 and x^2. If a product of factors equal 0, I feel that you should set each factor to 0 and solve. Especially when many students claim that 3=0 when x is -3
They might be thinking that way. But there's no evidence that anyone here did, and you can't read minds any better than I can.

And in saying "3x^2 [is] the product of two factors, namely 3 and x^2", are you denying that x is a factor? Of course not.

Yes, your way of saying it (which I sometimes do, too, when I first introduce the topic) is a way to emphasize that we are applying the "zero-factor theorem". However, I do that not because students don't see 3 as a factor, but because a few do, and get confused, thinking that 3 = 0 means x = 3. So I'm preemptively keeping them from going too far in that direction, and then tell them that they can ignore constant factors.

So a student should quickly move on from there and realize that they can just write 3x^2=0 and know that x = 0. So in teaching, go ahead and say what you did; but don't imply that it's wrong to say it in other ways.

One of the benefits of a forum like this is that different helpers may diagnose a student's problem in different ways, and something I don't think of, that you do, may be just what they need. Or it may not. In any case, it's not something to argue over. We're here to join forces in helping students, not to quibble over different styles.
 

Otis

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I'm finding it hard to believe that students would consider 0=3 as true. How can that happen?!

😟
 
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