Algebra help

Locust76

I've been doing math for a few hours now, so my brain is frying.


Solve for x.

This problem is kicking my ass for some reason. Can someone explain to me how I'm to go about solving this problem?

SeñorAmor

If memory serves, first you should add 3/x to both sides. Then, you should square both sides to remove the radical. Then it's much easier to solve for x.

Septus

Blah, it's been way too long for me. I got stuck at x= (1/4 + 9/x^2)(x-1)

VeritasVR

Solved.


This is my best guess. I'm not a math major and it's been a long time since I did non-intuituve algebra. Also, I know my scanner sucks balls.

But there you go. Enjoy.

GoodOmens

Big problem with Veritas's answer. When squaring the left side, you don't square each part sepearately. In other words, the square of the sum is not the sum of the squares.

Personally, I would start by multiplying the whole thing by x, then adding 3 to both sides. That would end up with:

xsqrt(x/x-1)=1/2x+3

Then square both sides to get:

(x^2)(x/x-1)=x^2/4+3x+9

Cleaning up the left side:

x^3/x-1=x^2/4+3x+9

Multiply both sides by x-1:

x^3=(1/4)x^3-(11/4)x^2+6x-9

Mutliply by 4:

4x^3=x^3-11x^2-24x-36

0=-3x^3-11x^2-24x-36

Then God help to factoring it.

Graphing this sucker gives you approximate solutions of 1.1, -2.2, and 4.8.
The -2.2 is wonky, though, when you put it into the original equation, and I can't tell why. Of course, approximate solutions may not be good enough for your teacher, in which case I can't help you further...this thing doesn't seem factorable.

supertall

The squareroot of x/x-1 isn't a trigonometric identity is it?

SeñorAmor

I got

3x^3 - 11x^2 - 24x = -36

And that's as far as I'm taking it.

Folken Fanel

SeñorAmor wrote:

I got

3x^3 - 11x^2 - 24x = -36

And that's as far as I'm taking it.

Yes this is absolutely correct. I dont know of any ways to solve for x analytically. I will post a graph of the function and the approximations for the function's roots in a few minutes...

Folken Fanel

Ok, after some kung-fu algebra you should get

0 = 3(x^3) - 11(x^2) - 24x + 36

You can let this polynomial be a function and graph it in a program like a graphing calculator, or something like Mathematica or MATLAB. The function looks like this:


Clearly, this function equals zero at three places.

The function crosses zero at:
x = -2.25198
x = 1.10758
x = 4.81106

homeobocks

Folken Fanel wrote:

Yes this is absolutely correct. I dont know of any ways to solve for x analytically.

The cubic equation. But it's a pain.

http://en.wikipedia.org/wiki/Cubic_function#Root-finding_formula

Big Dookie

Normally you would be able to find at least one of the roots of 0 = 3(x^3) - 11(x^2) - 24x + 36 using synthetic division, and then you can use the quadratic forumla or complete the square or whatever to find the other two. However, it appears to be impossible to find the first root using the integer factors of the constant term (believe me, I tried).

However, looking at the solution you came up with, that doesn't surprise me now, since none of them are integers. With that in mind, I don't know how you could find one of those roots on paper in order to find the other two. The only way you could solve for x would be to use a calculator or computer program. Unless there's some other way of doing it that I'm forgetting, that is.

Edit - Oh, right, the cubic equation. Yeah, screw that.

scrivenerjones

danger! the reason that people have been getting a weird negative answer is because squaring everything hides the fact that the sqrt[1/(x-1)] expression gives rise to badness when 0<x≤1. namely, the function is undefined at x=0, and involves taking the square root of a negative number up until x>1.

so the relevant part of the plot looks like this:


zoomed out, it looks like this:


note the discontinuity. so the positive solutions are right; the negative one isn't.

VeritasVR

Oops to the sum of squares. Actually the rest of mine works with the new values if you just make

sqrt(x/(x-1)) = A and 3/x = B, then do

(A - ^2 = (A^2 -2AB + B^2) or something,

then sub back in for A and B.

Locust76

My professor is a fucking moron. I asked him about this problem, and apparently there's not supposed to be a sqrt on x/(x-1), ALTHOUGH he typed { x/(x – 1)} in the final exam.doc, and {} was earlier in the course identified as the fucking MS Word equivalent of the sqrt function. (this is a class over the internet, btw)

God help me if I ever see this guy walking down the street.

Thanks for your outstanding analyses of a fucked up math problem

EDIT: Here's the solution that I came up with. using the CORRECT problem and not the fucked-up one the professor erroneously typed:


By the way, I ran this through Derive 6 and got the same results, so I know my answer is right... is there a way to get Derive 6 to actually SHOW the steps? Whenever I use the SOLVE function and click the "step" button, it always goes straight to the answer without explaining how it got there. Maybe the step button only works on more complicated shit? And I can't get Mathematica to do shit... I must not be using the thing right...

GoodOmens

Locust76 wrote:

My professor is a fucking moron. I asked him about this problem, and apparently there's not supposed to be a sqrt on x/(x-1), ALTHOUGH he typed { x/(x – 1)} in the final exam.doc, and {} was earlier in the course identified as the fucking MS Word equivalent of the sqrt function. (this is a class over the internet, btw)

Oh. Well. That's much easier then.

I'll get some strange looks tomorrow; I worked on this on the whiteboard of my classroom during PTA meetings last night, and left the work there until this morning so that I could figure out why -2.2 was not working. It'll be a Good Will Hunting Moment.

And, scrivnerjones, I suspected that was the problem, except that if you punch -2.2 into the entire radicand, it's positive (-2.2/-3.2), so that shouldn't matter.

Hirocon

The problem with the proposed solution -2.2 is that the steps leading to that solution are not reversible.

Say we have

a = b.

From this we know

a^2 = b^2.

BUT, from a^2 = b^2, we do NOT know that a = b. That step is not reversible. There is a solution to the equation a^2 = b^2 which does not solve the original equation a = b (namely, a = -b). -2.2 is that sort of solution. If you plug -2.2 into the original equation, but take the negative square root instead of the positive square root, the equation is balanced.

GoodOmens

Yep. That's it. I was being too stubborn about the solution. Just like I tell my students not to.

Folken Fanel

Locust76 wrote:

My professor is a fucking moron. I asked him about this problem, and apparently there's not supposed to be a sqrt on x/(x-1), ALTHOUGH he typed { x/(x – 1)} in the final exam.doc, and {} was earlier in the course identified as the fucking MS Word equivalent of the sqrt function. (this is a class over the internet, btw)

God help me if I ever see this guy walking down the street.

Thanks for your outstanding analyses of a fucked up math problem

EDIT: Here's the solution that I came up with. using the CORRECT problem and not the fucked-up one the professor erroneously typed:


By the way, I ran this through Derive 6 and got the same results, so I know my answer is right... is there a way to get Derive 6 to actually SHOW the steps? Whenever I use the SOLVE function and click the "step" button, it always goes straight to the answer without explaining how it got there. Maybe the step button only works on more complicated shit? And I can't get Mathematica to do shit... I must not be using the thing right...

Mathematica is a cruel mistress indeed. Its very picky about syntax. Didn't capitalize a sine function? Mathematica doesn't recognize it. Don't use square brackets when you should? Mathematica doesn't recognize it. The graph I posted, along with my solutions was found using mathematica, and it took me longer than it really should because Mathematica is so picky about syntax.

Nitsuj82

It's times like these that I'm glad I chose to major in business.

//And yet at the same time, I think I'd have more fun as the civil engineer I wanted to be.

Delzhand

The original equation in the OP has the solutions 1.10758, 4.81106.

Go go TI-83!