@Sim (you like Wayne’s world too? :)): Great question. I think the reverse FOIL thing comes down to pattern recognition. Realistically, we usually do manual factorization on nice, premade homework problems that have nice clean solutions. In the real world we let computers factorize for us.
After doing enough problems, you start to see that factorizations work out like this:
$$(x + a)(x + b) = x^2 + ax + bx + ab$$
Notice that the a*b term has no “x’s” involved. So going backwards, if you see a regular number by itself (like -15, in x^2 + 2x - 15), you start wondering what “a” and “b” could result in -15 (remember, a and b can be negative too):
-1 and 15
-15 and 1
-3 and 5
-5 and 3
The next step is to find the combination that has a difference of 2 (since we need a 2x term left over). In this case, it’s +5 and -3. So we get
That’s the process that goes into my head. It is brute force to some extent, and we can use the quadratic formula to blast through any equation automatically.
Honestly, most homework problems will be set up nicely so you can recognize the factors pretty quickly [what a and b have the difference we need?].
As usual a very thoughtful post. I’ll spend the week thinking about how this relates to my own sense that the reason to find each “component” of an equation is to figure out relative maxima and minima. Figuring out the “roots” of an equation is not anywhere near as interesting as figuring out the roots of its derivative. Is there an equation to track the error in my thinking?
Body is solid and is made up of magnesium alloy which makes this camera durable to last longer.
Mega pixels are how manufacturers measure the pixel count
of an image created by a camera. Macro is for getting those awesome close-up shots that are often seen on
sites like Flickr.
I like the “error” analogy very much. For years I always wonder why there is a error
function in a PDE. After your “system” analogy , I just remember that PDF used to discribe the control system, and the goal is minimum the error function. Though I don’t know whether there is someway to “factor” a PDE system .
It does appear, however, that hypnosis can help a person achieve his or her weight loss goals.
Avocados- Although not my favorite, are high
in fats, the good ones. Expect to lose all that extra flabbiness as you melt
off all your unwanted fat.
I tend towards very visual solutions when solving algebraic equations. Systems of equations makes sense to me because the process of solving boils down to finding out where two lines intersect, if they intersect at all.
y(1) = 6
y(2) = $$x^2 + x = x * (x + 1)$$
I know the parabola of y(2) has roots of 0 and -1. I also know that the parabola is increasing at equal rates about the line x=-0.5. Those two points where the parabola meets y=6? The ‘negative point’ of the parabola (left of x=-1) is a touch more negative than the ‘positive point’ (right of x=0). At least, if you’re drawing everything to scale. x = (-3, 2)? Sure, I’ll buy that answer. It fits the prior estimate.
Why do I do this? Because as the functions get more varied and complicated in these equations ($$2/x = 5x + 3$$), I can still graph ($$y = 5x(x + .6)$$; $$y = 2$$) on the back of an envelope and get a good idea of what the solution set should look like.
I admit this does have the drawback of making this equation 2D, and some people will wonder if they need to use the y coordinate. I call that a learning opportunity.
I hope this is related. Lately I was troubled about this very question: why is that the zeros of one parabola $$y=x^2+x-6$$ happen to be the very solution we need for $$x^2+x=6$$? Then is dawned on me that $$y=x^2+x-6$$ is really the composite function when we take the difference of $$y=x^2+x$$ and $$y=6$$. And the very places where this new function end up being zero is where the former two functions had exactly the same value. ie.: the very places where the former two functions intersected! This might be completely obvious to most people, but for me it was a new little insight.
You are super intelligent. I have been studying, and waiting for an “aha moment” for a long time, and it never came. I am glad that I stumbled onto your site, as it helps me visualize WHAT I am studying. I mean, it’s fine to memorize the unit circle, but what do all those values mean? I love seeing the practical uses for math - it is around us everywhere, in everyday life! My mom said I would never need higher math, but every aspect of life is structured around mathematical principles.
I am trying to help my daughter with factoring and she asks me what is the use of it.
I think that if I gave her your explanation she would be even more confused as I am.
I truely cant think of a reason for factoring as an every day use such as knowing your times tables. I learned it at school and cannot think of a single instance where Ive thought ah - factoring is the answer and I was a computer programmer for 30 years.
I did however come across a method for factoring on the internet where for example: 2x(squared) + 12x + 10
you multiply 2 x 10 = 20
get the possible factors for 20 ie. 1 x 20; 2 x 10; 4 x 5
the factors added together should = 12
therefore 2 and 10 are the factors
then replace 12x by 2x and 10x
thus: 2x(squared) + 2x + 10x + 10
then: 2x(x + 1) + 10(x+ 1)
then: (x + 1)(2x + 10) are the factors
Hi Clive, good question. It’s hard to justify most learning based on the test of “Will I use this every day?”. How often do we write by hand, count past 100, or name the planets?
A better approach, I think, is this: If you want to be really, really good at something, math gets you there. A civilization where people only know their times tables doesn’t build rockets, computers, telephones, cure diseases, etc. Now, we don’t need these awesome things: we can quietly plant vegetables and live in caves. But math is a superpower that pushes civilization forward.
Specifically: algebra is a way to write down relationships, or to model a scenario. Factoring helps you simplify those relationships so you can better understand them (figure out when they reach a certain value, or work backwards from a value to what inputs got you there). You can manually trial-and-error to factor, but it breaks down for more difficult scenarios like 2x(squared) + 4x + 7.
hi, I have been in this site since yesteday, I am hooked, I was particularly interested in negativ numbers , what do u think computer programming analogs to math, math is best seen as a programming exeecise
Hi Kalid, your site has been a big source of hope and inspiration to me. I study it almost on daily basis. Can you pls help me understand modelling in mathematical language am stuckstuck. Thanks