Topic, Offtopic post

  Sketching, writing, marking, erasing, colouring, measuring, cutting, and folding.

  Pencils, pens, markers, erasers, crayons, rulers, scissors and paper.

  That was a brilliant method Nelson, I never really thought of it that way. I forgot the term polynomial and square root in terms of solving equations. There are many different solutions and possibilities.

  @nelson Nice solution! I tried to reverse engineer my way and it will involve guessing solutions anyway so I think your way is better. That was the way I expected you to solve that question. I came up with it by the method in the spoiler at the bottom (because it reveals a way to solve it).

As for the general procedure for solving a quartic, I think that’s a little crazy to suggest a problem that requires it, because the solutions are far too large to write out. I used to think that the ‘guess and factor/divide’ methods weren’t as direct but honestly solving that with a general procedure is far too extreme! As far as I understand though it would require solving a general cubic, which itself isn’t commonly known how to do.

@chris Yes, I have heard of inequalities. Linear inequalities are usually almost as simple as linear equations but adding higher powers or other functions makes inequalities far more tricky and require a lot of work.

To generate the problem:

  Have you guys heard of inequalities? It's one of the algebra terms back in high school when I attended in 2009.

  equation x^4-2x^3-6x^2+10x-3=0

The general method to solve this kind of equation is very difficult. I don't remember the whole process but it seems to me that it will be necessary to solve an equation of degree 3.

The simplest method is to find obvious solutions and factor the polynomial:

  Pumpkins, Scarecrows, Ravens, Skeletons, Witches and Goblins

  We're going to the pumpkin patch tonight for a Halloween party. Enjoy this holiday guys. You can get candy from trick-or-treaters.

  Pi, 3.14, #, circumference, radius and circles

  Happy Halloween everyone. Enjoy all the decorations and candy that's passed out by trick-or-treaters. However I'm too old to do that sort of thing now.

  Here’s a potentially trickier to solve equation if you want something even more difficult. I know of two methods that will work here, but it may not be easy to ‘guess’ one method, the other is similar to Nelson’s other solution - which is the typical way to solve this type of equation:

x^4-2x^3-6x^2+10x-3=0

(Again there are four separate solutions!)

  s^2, pi*r^2, pi*r^2*h, s^3, b*h/2, (3*sqrt3/2)*s^2 (regular).

  Squares, Circles, Cylinders, Cubes, Triangles and Hexagons

  Red Riding Hood, Three Little Pigs and the Big Bad Wolf.

  , ,

  4, 8, 12, 16 and 20

  0, 1, e, pi, i

  Grass, dirt, sand, water, ice.

  Rain, thunder, wind, hail, tornados and hurricanes.

  @nelson Very good! Polynomial division is definitely a possible alternative to my substitution, and honestly I didn't think the equation was too hard when I first set it, I guess I should have assumed most people didn't see it.

@suhangha I derived the formulae (there are technically two if you count the + and - versions as separate though for legitimate triangular numbers you should always use the + formula), they are very interesting and can provide a generalisation to the question - 'If 2, -1/4, or whatever were a triangular number, what place would it be?'

I'd be interested to see if I got the correct answer though, I checked it and it seems to work.

  @CG52: no, your equation wasn't too difficult, it's just that I didn't see it. I remember learning how to solve this kind of equation over 50 years ago :)

The general way to solve it is as you mentioned:


second solution:

  Anyone interested in the inverse formula of the Triangular Number?

I figured out the formula a while ago on my own.

  So, what are you all going to do this weekend? I might go to the mall tomorrow sometime.

  c=10

Most standard method:


Maybe my equation was too difficult…

Hint:

  Now solve for c.

10c+8=108

This will be an easy one for you guys.

  The equation x^2-(a+b)x+ab=0 will have the solutions x=a and x=b, that's how you can make a polynomial equation with 2 solutions. More than 2 requires higher order polynomials. There are other ways, like trigonometric equations but that might not be familiar to you anymore if you stopped studying mathematics many years ago.

  Well done with those methods CG52, now I proved to make an equation with more than 1 solution.

  a=4

Method 1 - Normal:


Method 2 - Factoring:


A bit harder, try to solve for x:

x^4-5x^2+4=0 (there are 4 solutions!)

  Solve for a.

2a-8=16-4a

  I don’t actually know where the word ‘transcendental’ comes from. It’s probably to do with the numbers ‘transcending’ the algebraic methods of constructing numbers. Similarly, irrational numbers are those that aren’t ratios of whole numbers (so can’t be represented as fractions).

Inequalities can be confusing, particularly if it is a rational function instead of a polynomial, but even a simple quadratic can be trickier than you might think.

Mathematics takes a lot of practice to learn and while some people are naturally gifted at it, I do think that 90-95% of people can understand maths to a level that would surprise themselves. Of course I understand that this stuff is very abstract for a lot of people without the background knowledge or the desire to learn it xD. Even simpler stuff though, practice is the only way you can get better, but when you do a lot of those problems, improvements will naturally occur.