Math Is Fun Forum

harpazo1965 wrote:

Please copy and paste the following link. Tell me if you can see the image.

https://imgur.com/gallery/PXM8kTT

Yes, I can...as part of the following Imgur web page (a screen capture):

https://i.imgur.com/n3c7hgA.png

However, while your link works for me now, I've found from your other posts that your web page links only work for a limited time, if at all.

hi imcute

As you are new you perhaps won't have been following the difficulty here over a number of threads.  On a previous server it was posssible to upload images directly to MIF but difficulties with that server meant the forum had to swiitch over and the direct upload facility was lost.

I use Imgur to get images here using a bcc code address from Imgur.  I advised harpazo1965 to do the same.  He signed up and uploaded his image there but the site does not support bcc on their mobile app, so this hasn't helped him.

If you know a way to get an Imgur image into bcc format on a mobile we'd all be really pleased to know it.  Or maybe you know another way for him to get images here.  Thanks.

Bob

I am trying Phrontister's method for creating an Imgur image.

step 1. On my mobile, log in to Imgur and select an image.

step 2. Bottom right of screen, click the arrow and choose copy URL.

This is that url.

https://imgur.com/a/BURsC8w

If I enter it into my laptop browser address bar I can view the image. I'm not logged in to Imgur on my laptop so I assume anyone can do this.

step 3. On my laptop right click the image and choose 'copy image link'

step 4.  I then pasted that back into this post. Here is the link:

https://i.imgur.com/7RjDiAX.jpeg

Note. It is different to the above and has jpeg as an extension.

step 5. Add bcc code to make the image appear. open square bracket, img close square bracket; image link address;open square bracket /img close bracket.  Here is the resulting image.

https://i.imgur.com/7RjDiAX.jpeg

So, if a member is limited to a mobile I can convert their image so it appears in their post. They would have to do steps 1 and 2, and I'd have to do the rest.

Remaining question. Can steps 3 - 5 be done on a mobile?  Still working on that.

Bob

2x^2 + (432/49x) = 3

That should work but I still think the graph is the key. Once you have that the algebra is easy.

Bob

It's got a vertical asymptote at x=3. And a horizontal one at y=1.

If you attempt to solve for y=1 you get no x value.

So that's a clue. Try to construct the graph

Bob

The equation

has two "obvious" solutions: x=2 and x=4. They are "obvious" because they are relatively easy to find by sheer guessing or by sketching the graphs of the two functions involved, and once guessed they are trivial to verify.

https://useruploads.socratic.org/apnbo8bVTC6ioBCx7wzf_AutoGraph.jpg

There is a third solution with x<0. That this solution exists is fairly obvious as well: just consider the general behavior of the two functions for negative values of x and you will see that they must intersect at least once - in fact it's not hard to see that they intersect exactly once in that region.

However, it is not possible (we believe!) to write down an explicit formula for that solution by using rational numbers, arithmetic operations, and the so-called Elementary Functions: exponentials, logarithms and trigonometric functions. It is possible to write down such an explicit expression using a decidedly less standard function, the Lambert W function, and that's pretty straightforward too.

Proving the negative assertion above is far from trivial, and in fact I don't think that such a proof is known. Just because the solution can be expressed by evaluating W at ln(2)/2 (and making a few other simple calculations) doesn't rule out the possibility that there is also another expression for that same solution using elementary functions evaluated at rational numbers.

There's a brilliant and readable paper by Tim Chow on this topic, called "What is a closed form number?" . I highly recommend it to anyone interested in gaining a deeper understanding of those questions. It is a little surprising that many people wouldn't hesitate to make the assertion I made above, about the negative solution being impossible to write down "in closed form", without having proof of that fact and often without even being able to express precisely what the claim is.

We can see there is one solution in the interval -1 < x < 0 and a second solution at what appears to be x = 2 and a third at what appears to be x = 4.  The two solution x = 2, 4  can easily be verified to be exact by substitution.

We can find the third solution numerically, using Newton-Rhapson method

Let

and using the Newton-Rhapson method we use the following iterative sequence.

And we conclude that the remaining solution is

to 4 dp.

Many thanks, Phro.  That's just what I needed. I'm gong to take out your hide tags to see if the image shows as was wanted.  If so, you have found a way to get the image onto MIF from imgur. Yay! It works!

As to the poster's question its simply this.  All x lies between minus infinity and plus infinity so why waste space including an inequality.

Bob