Hatrack River Forum: I can't take it anymore! A bit of help on this math problem? *SOLVED*

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Posted by Vadon (Member # 4561) on :

I've been working on this one math problem for the past two weeks. I've redone all the other ones that I have gotten wrong, except this one... I just can't find any other answer. (I am allowed to ask for help.)

3X^2 + 27 = 0

I'm supposed to simplify that as far as I can, but all I ever get is +/- 3i (Which she wrote as wrong.)... Can someone else find something different?

[ January 03, 2005, 07:26 PM: Message edited by: Vadon ]

Posted by Ryan Hart (Member # 5513) on :

+/- 9i

3(9i)2+27=0

Posted by Vadon (Member # 4561) on :

Hey... that does work... Make the 9i negative from the squaring... then multiply by 3, you get -27... 27-27 =0... 0=0... that works... Thanks. ^_^

Hm, actually... um... 9i^2 = -81 doesn't it?

[ January 02, 2005, 06:36 PM: Message edited by: Vadon ]

Posted by JonnyNotSoBravo (Member # 5715) on :

It's +/-3i not 9i....-27/3=-9, and the sq. rt of -9 is +/-3i....

Posted by Vadon (Member # 4561) on :

That's what I get... but she marked it wrong... I'm starting to wonder if she did it wrong herself...

Posted by JonnyNotSoBravo (Member # 5715) on :

Perhaps you aren't providing enough information for us to help you solve this. Was this a story problem? Are there units? Was this the problem that was assigned? Is it acceptable to use imaginary numbers? Were there extra parentheses you didn't see? Did you copy down the problem correctly?

I used to believe the teacher was wrong, too, but it's just fairly improbable that she is, so consider other options before possibly embarassing yourself.

Posted by Vadon (Member # 4561) on :

You'd be surprised, she does get problems wrong at times...

This is the problem exactly as it is shown... We're supposed to solve for X and it is acceptable to use imaginairy numbers.

Posted by Icarus (Member # 3162) on :

Teachers certainly make mistakes, and it's not a mark of incompetence that they do so. Can you ask the teacher why she marked it wrong?

Posted by Paul Goldner (Member # 1910) on :

(+/-3)i is the correct answer.

Posted by ssywak (Member # 807) on :

And you're positive that it's NOT

3*X^2 - 27 = 0

Right?

(My stupid question for the day)

Posted by Vadon (Member # 4561) on :

This is for a corrective, I can re-do problems I got wrong as long as I show how I did it, but what I answered with it was what you apparantly are getting as well. +/-3i.

She isn't incompetent, she is one of the best math teachers I've had, she just... makes mistakes at times. ^_^

I'm posititve, but... is there a difference between 3X^2 and 3*X^2?

It's just um... one buggy problem, I'm almost positive that it's +/- 3i, but... alas that she marked wrong.

Posted by Boris (Member # 6935) on :

I suck at math, but here goes...
3X^2 +27 = 0...
So it's safe to say this...
3x^2 = -27
Right?
so then,
(x^2)/3 = -27/3
x^2 = -9
x = (Square root of -9)
I don't remember anything about imaginaries...so it would be safe to say that
x = -3i from that, right?
I don't know, you might be able to get something out of that, though. I'm just an english major [Smile]

Oh wait, it'd be just -3i, not +/-3i. Wouldn't it? I mean, you have the 3x^2 part, but the constant is going to be negative when you move it over. something can only by +/- if the resultant constant is positive right?

[ January 03, 2005, 12:31 AM: Message edited by: Boris ]

Posted by dkw (Member # 3264) on :

quote:
I'm posititve, but... is there a difference between 3X^2 and 3*X^2?

Yes.

Edit, assuming you mean a difference between (3x)^2 and 3(x)^2.

[ January 03, 2005, 12:44 AM: Message edited by: dkw ]

Posted by punwit (Member # 6388) on :

It's been a long time since I've done equations like this so I could be wrong but why would the answer be (+/-) 3i? Why the +/- instead of just -3i?

Realized I just echoed Boris.

[ January 03, 2005, 08:47 AM: Message edited by: punwit ]

Posted by Choobak (Member # 7083) on :

3X^2 + 27 = 0

Ah ! les joies des mathématiques !

When I study, my profs said you must know "par coeur" the types of solutions of the equations :

X² + w² = 0 -> 2 solutions : iw and -iw
X² - w² = 0 -> 2 solutions : w and -w

So here, you have :

3X²+ 27 = 0
<=> X² + 9 = 0
<=> X² + 3² = 0
<=> X = 3i or X = -3i

CQFD !

Good courage !

[ January 03, 2005, 09:02 AM: Message edited by: Choobak ]

Posted by punwit (Member # 6388) on :

I did a bit of research and I think I had it backwards. It's been so long and I didn't realize that i^2 =-1. Given that it would follow that:
3x^2+27 = 0
3x^2 = -27
x^2 = -9
If i^2 = -1 then x=3i

I just don't see how the answer could be -3i since i^2 = -1. Then your equation would read (-3(-1))^2 = -9 and in my book that would equate to 9 = -9

[ January 03, 2005, 09:40 AM: Message edited by: punwit ]

Posted by Choobak (Member # 7083) on :

wrong !

X is a pure imaginary. explanation :

you have X²=-9
<=> X² = -1*9
<=> X = rootsquare(-1*9) or X = - rootsquare(-1*9) (logical !, i'll use now the +/- symbol, ok ?)

<=> X = +/- rootsquare(-1)*rootsquare(9)
however rootsquare(-1) = i because 1² = -1
and rootsquare(9)= 3 (no problem here)

so X = +/- i*3 -> two complexe solutions !

Understand ?

Posted by punwit (Member # 6388) on :

I'm having a hard time getting my head around this. It's been along time since I practiced any of these skills and you are obviously more skilled than I am. I've done some more digging and thought I'd link this page that seems to support my position. They solve for the sqrt of -9 and get an answer of 3i which is exactly what our problem resolves to.

Posted by Dagonee (Member # 5818) on :

Yeah, I'm thinking it's the +/- which is wrong. But I haven't taken math since 1992.

Posted by fugu13 (Member # 2859) on :

punwit --

3i is just 3 times i.

-3i is just -3 times i.

(3i)^2 is 3 times i times 3 times i, or 3 times 3 times i times i, or 9 times -1, or -9.

(-3i)^2 is -3 times i times -3 times i, or -3 times -3 times i times i, or 9 times -1, or -9.

since (-3i)^2 = -9, at least one square root of -9 must be -3i. Ditto for 3i, of course.

The answer is definitely +/- 3i for the problem as stated.

If nothing else you should be able to plug the equation into a TI-92 or 89 if the teacher doesn't believe you, its sort of hard to deny a computed answer for a simple problem like that.

Posted by Choobak (Member # 7083) on :

Ok ! I understand where is your Uncomprehension.

It's not the fact that is a complexe solution. You forget that when you make a number at square (X²) the sign become +.

Explanation : (-2)² = 4 because you can write :
(-2)² = (-2)*(-2) however -*- = +

But 2² is also equal to 4.

So When You want the rootsquare of 4, you have two answers : 2 and -2 (2² = 4 and same for (-2)²=4)

Here, it is the same problem : you have X² = -9

So if X = -3i, X² = (-3i)²
<=> X² = (-3)²*i² Ok ? I continue :
<=> X² = 9*(-1)
<=> X² = -9
I can conclude X = -3i is a solution.

If X = 3i, X² =(3i)²
<=> X² = 9*(-1)
<=> X² = -9
So X = 3i is also a solution.

Don't forget there are 2 solutions for a Square !

To introduce my maths capacity, I am Engineer, and study many and many years Maths (and more : Physics, Mécanic, Electronic...).

Be entrusting about what i wrote. [Wink]

Posted by punwit (Member # 6388) on :

I suppose I can follow that explanation but to me it only makes sense if you notate it as (+/-3)i

Posted by punwit (Member # 6388) on :

I wasn't disputing you so much as trying to find an answer that made sense to me. It's been over 20 yrs since I had much reason to think like this. I can wrap my head around the answer now and I appreciate the mental stimulation I've received this morning.

Posted by fugu13 (Member # 2859) on :

The unary "operator" +/- has a very high (right hand) association in mathematical notation, which means in pretty much all cases it will apply to the smallest expression most directly to the left (that is, the smallest expression one can make and still have it be directly to the left).

Now, there is also a binary operator +/- (plus or minus instead of positive or negative as in the unary version), which works at about the same level as, well, plus or minus.

However, in this case it doesn't matter at all, as -3i = (-3)i = -(3i) = (-i)3 = 3(-i) = -(i3), et cetera.

Similarly, +/-3i = (+/-3)i = (+/-i)3, et cetera.

Posted by Choobak (Member # 7083) on :

Happy to help you. Ask again if you have a problem. All Hatrack is here to help you. [Wave]

Posted by aspectre (Member # 2222) on :

If "...she is one of the best math teachers..", then the solution is a complex number:
ie the solution does not lie on the real number line but rather on the complex number plane.
And so the solution must be of dimension 2:
ie must be presented in the form (a + bi), in which a and b are real numbers,
with a representing the real component and bi representing the imaginary component.

And since you are taking the square root, there must be two points on the complex plane which satisfy the conditions of the equation.
In this case, a = 0, and b = +3 and b = -3.
And so, X = 0 + 3i and X = 0 - 3i;
or X = (0 + 3i) , (0 - 3i), or even possibly X = (0 +/- 3i)
(you'll hafta ask your teacher about which is/are acceptable);
remember that you are describing points on a complex plane.

However, you stated both "I'm supposed to simplify that as far as I can" and "We're supposed to solve for X",
which are not necessarily compatible with each other:
eg one wouldn't simplify Y^2 + 1 into (Y + i)(Y - i).
Again, you'll hafta ask your teacher about what she wants.

And no, your teacher isn't just being nitpicky or tricky.
Mathematics is a formalism; and teaching the proper form is part of a math instructor's job.

[ January 03, 2005, 03:26 PM: Message edited by: aspectre ]

Posted by Jonathan Howard (Member # 6934) on :

I may be only 14, but I know one thing.

The square root of a number is ALWAYS the positive.

SQRT of 4 = 2. Not -2, nor anything else. 2.
2^2 = 4. -2^2 = 4. SQRT 4 = 2 ONLY.

I told my maths teacher that this is pathetic! You might as well define a SQRT in a better way; one that would make SQRT 4 = |2|. But he said that the definition was simply made to accept the positives only, and that the subsequent negatives can be implied from the answer.

Thus:

3X^2+27=0
3X^2=-27
X^2=-9
X=SQRT: -9
X=3i

not +/-3i, not -3i. 3i, and that's that. Tell her that Nethanel Altschuler would solve that way *wink*.

Jonny

Posted by fugu13 (Member # 2859) on :

No offense, but you certainly are only 14. The square root of a number is any number which, when squared, returns the number we are taking the root of. 4 has 2 square roots. Perhaps your math teacher considers it sufficient to list just the positive root, but the negative root most certainly exists. And I rather suspect I have had more math than your math professor (though I could be wrong).

However, don't take my word, take the word of a modern mathematical genius (well, of his web site):

http://mathworld.wolfram.com/SquareRoot.html

Regarding him talking about the principle and other square roots, he was not asked to find the square root(s), he was asked to find the solution. The solution to the equation is any value(s) that, when substituted in the equation, result in truth.

In this case taking the square root is merely a means we have used to arrive at the solutions. I shall hereby demonstrate that both 3i and -3i are solutions (as someone has doubtlessly done earlier, but it makes this post nicely complete).

code:

3*(3i)^2 + 27 = 0

3*3i*3i + 27 = 0

3*3*3*i*i + 27 = 0

27*-1 + 27 = 0

-27 + 27 = 0

0 = 0

True

code:

3*(-3i)^2 + 27 = 0

3*(-3i)*(-3i) + 27 = 0

3 * (-3) * (-3) * i * i + 27 = 0

3 * 9 * -1 + 27 = 0

-27 + 27 = 0

0 = 0

True

Posted by Icarus (Member # 3162) on :

Occam's Razor, folks.

Either the teacher marked it wrong by mistake, or there is something we have not been told.

-o-

fugu, are you a grad student in math?

Posted by fugu13 (Member # 2859) on :

No, though I was an undergrad math major for a couple of years. Its been my experience that under the level of high school Calculus most high school math teachers haven't had much math beyond a couple semesters of Calculus in college, something I surpassed by the end of my junior year in high school.

Its quite possible his math teacher has more training, but I'd bet its unlikely.

Posted by fugu13 (Member # 2859) on :

Yes, Ic, that's what I'm betting, though I must say I'm amazed at how hard it is to convince people of the correct answer to a simple Algebra problem (this is not a jab at those who haven't done math in a while at all, or at anyone, just an observation).

Are you sure, Vadon, that there was no additional context for this problem? Perhaps a statement at the beginning of the problem set of some kind?

Posted by punwit (Member # 6388) on :

fugu13 I have to agree with ya. I was having a hard time because I was being dense. Once I figured out what I was doing wrong I had to get an ice pack for the headache caused by hand to forehead action.

Posted by Vadon (Member # 4561) on :

Ok, you may now all beat me with many large wooden spoons. Turns out the answer I turned into her to get it wrong was missing one crucial component... the +/-... yeah. So, thanks for your help. ^_^

I forgot that when taking the square root of a square number there is always the +/- when I did the problem. Ah, curse those times of memory lapse...

But, the answer is... *trumpets blare* +/-3i!

Posted by Sara Sasse (Member # 6804) on :

Yay, problem solved. [Smile]

(Choobak, you are too cute. [Wave]

)

Posted by Icarus (Member # 3162) on :

quote:
No, though I was an undergrad math major for a couple of years. Its been my experience that under the level of high school Calculus most high school math teachers haven't had much math beyond a couple semesters of Calculus in college, something I surpassed by the end of my junior year in high school.

Its quite possible his math teacher has more training, but I'd bet its unlikely.

You know, my first instinct was to be irritated by this statement . . . except it's fairly accurate. :-\

I wouldn't go quite so far as to say most high school math teachers are so underqualified, though a lot of them are. I wonder if there are numbers on such things.

I would temper that statement . . . I wouldn't say it's unlikely, necessarily . . . I dunno; I just hate the thought that this could be anybody's off-the-cuff assumption about me without knowing anything else about me other than the fact that I teach high school math. :-\

Posted by Dagonee (Member # 5818) on :

The more pressing issue is to wonder why Wolfram and Hart are running a math info site. What are they planning? [Angst]

Posted by Sara Sasse (Member # 6804) on :

Exponential growth, Dag. [Smile]

quote:
I wonder if there are numbers on such things.

Posted by Vadon (Member # 4561) on :

I wouldn't doubt that statement on many of the math teachers at my school. However, my math teacher... she just... knows how to do it. [Dont Know]

Posted by Icarus (Member # 3162) on :

When I taught middle school, I was almost always the only teacher with a degree in math. Since I've been teaching high school, I haven't been able to suss out that kind of information on any of my colleagues. I might conceivable still be the only math major, but this would surprise me.

-o-

Sara, [Big Grin]

Posted by Paul Goldner (Member # 1910) on :

At least at my high school, all the math teachers had undergrad degrees in math, and 5 had masters.

Posted by Icarus (Member # 3162) on :

quote:
all the math teachers had undergrad degrees in math,

How do you know? Were their qualifications posted somewhere?