An annoying quadratic differential

[chrisname]

So, I have a differentiation question where I have to find the differential of y = 3x³ + x² - 2x + 1
I got,
dy
-- = 9x² + 2x - 2
dx
All fine so far.

Now I have to find the co-ordinates of the two points where
dy
-- = 0
dx

so obviously I'm factorising a quadratic:
9x² + 2x - 2 = 0
But I can't figure out how I can factorise this.

I've tried:
(3x + 1)(3x - 2) = 9x² - 3x - 2
(9x + 1)(x - 2) = 9x² - 17x - 2
(x + 1)(9x - 2) = 9x² + 7x - 2

All else I can think of is using fractions/decimals but I don't really want to. I had a brief moment of hope when I looked at my textbook (thinking "I must've written this down wrong) and saw "2x³", but it was the previous question.

How do I factorise this quadratic? I can't use the formula because I'm not allowed a calculator in this module, and I can't remember how to complete the square. Factorising is usually the best way of solving these for me, but I can't see how it could possibly work.

Edit: Also, does this board have [latex] tags at all? They would be useful.

[helios]

Well, b^2-4*a*c=72, and sqrt(72) is an irrational, so there's no way you'll get it by guessing.

Just because you can't use a calculator doesn't mean you can't use the formula. Simply leave the result expressed in relation to a square root, which is what you should do anyway for irrational values.

[m4ster r0shi]

chrisname wrote:

Now I have to find the co-ordinates of the two points where
dy
-- = 0
dx

Why do you want to do this? You already have the differential. It is dy=(9x² + 2x - 2)dx.

[Albatross]

helios +1. If you end up with an irrational number and you can't use a calculator or your head to perform the necessary calculations to get it to a decimal form, just leave the square root, pi, e, whatever.

@r0shi: He seems to be finding critical points.

This board doesn't have latex tags. Sorry.

-Albatross

[chrisname]

@helios,
True, I could leave the answer as a surd. My teacher said not to do it as we haven't learned how to do that yet.

@m4ster r0shi,
The question asks you to find the two points where dx/dy = 0.

@Albatross,
Yeah. Shame about the LaTeX tags.

We're done with differentiation for now, we've started integration. I'm so glad I took maths, I really wanted to learn Calculus. It's actually quite enjoyable.

[helios]

True, I could leave the answer as a surd. My teacher said not to do it as we haven't learned how to do that yet.

So what, you have to leave the result approximated? That's preposterous! What kind of math course is this?

[chrisname]

No, by "it" I meant "the question". Not leaving answers as surds.

Edit: I'm waiting for helios to make 4 more posts...

[helios]

"My teacher said not to do the question"?

[chrisname]

Yeah.

Also nice quads.

[moorecm]

What's wrong with the quadratic formula?

[chrisname]

I didn't want to use it because I wanted to get solutions for x that were integers without using a calculator.

[helios]

Regardless of whether you were using a calculator, how would you have found that the solutions weren't integers without using the formula (other than eventually getting bored or running out of probable integer solutions)?

[moorecm]

Why does the solution have to be integral? What's wrong with solving it symbolically?

[kfmfe04]

+1 moorecm, helios

Almost all the time, closed-form solutions beats numerical methods, including "guessing with integers", etc, so use closed-form solutions whenever you can.

You will find many problems later in engineering and physics which will force you to use numerical methods - you are lucky this problem is not one of them. I'm an engineer by training so I'm ok with numerical solutions, but there are many pure mathematicians who totally despise solutions by numerical methods.

[tition]

Regardless of whether you were using a calculator, how would you have found that the solutions weren't integers without using the formula (other than eventually getting bored or running out of probable integer solutions)?

http://en.wikipedia.org/wiki/Rational_root_theorem

There is also a method due to Kroeneker to factorize polynomials over the rationals (if you take the 1/N in front of the bracket you will get the polynomials to look integral).


[Edit:] I do not know an algorithm to figure out whether a polynomial is solvable with radicals. (As you might have heard/know, polynomial equations of degree larger than [edit: or equal to] 5 do not have to have solutions using radicals (=square, triple, ..., n^th roots). In order to solve such polynomial equations you need to introduce new (transcedent) functions).

[helios]

I would have thought that engineers and their obsession with precision loved analytical solutions and symbolic computation.
Did I ever mention that Maxima is freaking awesome?

solve([9*x^2+2*x-2], [x]);

[x = -(sqrt(19)+1)/9, x = (sqrt(19)-1)/9]
(In the GUI, the results are formatted in mathematical notation.)

EDIT: If I didn't know about that, there was no way he could know about it.

[tition]

Hey, try SAGE!

It includes Maxima as a sub-package, so no problem with that! You can use it through the browser however, and program in Python from within!

[Edit]: and it uses Wikipedia-like way (via javascript) to render math formulas beautifully!

[helios]

I did, and I didn't like it. Run a virtual machine (for someone who hasn't tried SAGE, I'm talking about a full fledged x86 emulator, not something like the JVM) just to run a symbolic computation suite? Absurd! Maxima does everything I need with a fraction of the resources.

[tition]

Umm, does SAGE run a x86 emulator?

[kfmfe04]

Of course, engineers love analytical solutions, too! But, we won't turn our noses up at Monte Carlo methods or most any other solution that gives us a good answer (it doesn't even have to be the best answer). In the past, I have met mathematicians in my Financial Engineering classes who detested using Difference Methods, because the financial derivatives we were looking at were too complicated to have closed-form solutions.

Anyhow, my point was, the OP should be happy to get an answer by any method that he can!