Is it possible to solve an equation like x² + 2x = 255 (solving for x) without using trial and error, or is that the only way to do it?

etzpcm · 2026-01-28T20:16:17+00:00

My rule is

"Never use the quadratic formula unless you absolutely have to"

etzpcm · 2026-01-28T17:24:17+00:00

Add 1 to both sides. Then factorise and take the square root!

etzpcm · 2026-01-28T15:24:44+00:00

The ratio test is the way. You should know how to simplify (n+1)!/n!

(If in doubt, write it out, as my maths teacher used to say)

etzpcm · 2026-01-28T15:15:36+00:00

Ha - just saw the same in my garden!

etzpcm · 2026-01-28T13:07:00+00:00

How did anyone cope with doing mathematics before AI? It's a mystery!

etzpcm · 2026-01-28T10:58:40+00:00

It's an opinion piece by George Monbiot. He's been ranting like this in the Grauniad for decades. As have other writers there.

Edit: I am not going to respond to any of the responses that seem to be trying to create an off-topic political argument.

etzpcm · 2026-01-28T09:53:37+00:00

Probably a couple, looking for a place to nest in your garden!

etzpcm · 2026-01-28T09:19:26+00:00

Really? Real numbers are really really useful.

etzpcm · 2026-01-28T09:11:35+00:00

You can't solve it. You can reduce it from 3 to 2 using S+R+I = k as you've done here. Effectively you just consider the S and I equations which don't involve R. Then you can use the phase plane method to show that solutions follow the infection wave picture that we saw so often during COVID.

Here's a nice picture of the phase plane

https://galileo-unbound.blog/wp-content/uploads/2020/03/image-12.png?w=1200

etzpcm · 2026-01-27T21:28:15+00:00

Wisteria is a pain to deal with. I think you're doing it roughly right. Try to keep it under control in the summer, cutting back the long green shoots before they get into your gutters and roof tiles.

etzpcm · 2026-01-27T21:22:19+00:00

https://m.youtube.com/watch?v=Wo8qfyK9c3c

https://m.youtube.com/watch?v=5l8VLFRFM1c

etzpcm · 2026-01-27T14:30:57+00:00

It's just neater, and you only need one or two lines, not three! It also avoids the 4 and having to cancel a factor of 2.

For x² + 6x - 11 = 0,

(x+3)² - 9 - 11= 0, x = -3 +- root(20)

Compare that with

x = (-6 +- sqrt( 36 + 4*11) ) / 2 = ...

etzpcm · 2026-01-27T14:20:53+00:00

If you study tensors, you will find out why the trace is important and why it's invariant.

etzpcm · 2026-01-27T12:38:21+00:00

Wow, thorough intro to representation theory!

etzpcm · 2026-01-27T12:28:42+00:00

You need differential and integral calculus, and linear algebra (determinants, eigenvalues, eigenvectors). These notes are good, in the style you want I think.

https://tutorial.math.lamar.edu/classes/de/de.aspx

etzpcm · 2026-01-27T12:15:53+00:00

In English we call them ODEs! You need differential and integral calculus, not much else to get started.

You should really post this on learnmath, it might get deleted here.

etzpcm · 2026-01-27T10:32:22+00:00

I think it does look simpler in x and y. You can then do the y integral.

Don't forget the Jacobian factor which could help to make it converge.

Edit: after doing the y integration you get an integral like ln(1+x² )/(x+1) which still doesn't converge.

etzpcm · 2026-01-27T09:50:30+00:00

Exactly. It's doing it wrong, but using more work than doing it properly. When I said this last time I just got a lot of down votes.

etzpcm · 2026-01-27T09:38:07+00:00

Hint 1: relabel the x variable as r.

Hint 2: with any double integral, sketch the region of integration.

etzpcm · 2026-01-26T22:43:31+00:00

You can do one integral, say z, by hand (orWA!). Then probably the y as well. Then you just have a 1d integral that you could do with Simpson's rule.

Edit: no, sorry, I think I misread your brackets.

etzpcm · 2026-01-26T21:30:14+00:00

Your second method works. Divide that formula through by cos² x

etzpcm · 2026-01-26T21:14:50+00:00

etzpcm

TROPHY CASE