Yesterday, I witnessed something pretty cool and reminded me of my last technology class on voicethreads.  I was in between teaching classes, and the teacher I share a room with had just finished explaining/introducing the quadratic formula in his intermediate math class.  We were discussing the different songs we have used and learned to remember it by, some including theme songs to semame street and notre dame football.  He was trying to sing his to me, but claimed he was “tone deaf”, so he quickly logged onto his class website and pulled up a video!  It was of a teacher that he co-teaches with singing the actual song…..obviously with a better voice!  He put it on there to remind students of the quadratic formula and its significance, while keeping math fun and entertaining.  The kids will always have access to it, which is nice for those nights that they miss their math teachers so much that they have to watch them online

Program Description

Quadratic equation is an equation where the highest power of x is x².  The general form of a quadratic equation is

ax² + bx + c = 0; where a≠ 0.

The letters a, b, and c are called coefficients: a is the quadratic coefficient of x², b is linear coefficient of x, and c is the constant coefficient.

One of the methods to solve quadratic equation is the use of quadratic formula given below:

Here is a program that will ask for the three coefficients and will solve for the roots of x using the quadratic formula.

Output

Enter a value for coefficient a :  1

Enter a value for coefficient b :  4

Enter a value for coefficient c :  3

The roots of x are : -1 and 3

Source Code

#include <iostream>
#include <cmath>
using namespace std;

int main()
{
int a, b, c;
cout<<”Enter value for coefficient a: “;
cin>>a;
cout<<”Enter value for coefficient b: “;
cin>>b;
cout<<”Enter value for coefficient c: “;
cin>>c;
double in=pow(b,2.0)-(4*a*c);
float num1=-b+sqrt(in);
float num2=-b-sqrt(in);
float x1=num1/(2*a);
float x2=num2/(2*a);
cout<<”\n \n”;
cout<<”The roots of x are: “<<x1<<” and “<<x2<<endl;
return 0;
}

With my freshly minted PrimeOracle, I’m ready to attack problem 27, in which we are asked to find polynomials that generate long strings of consecutive primes starting when you plug in 0.

Basically my idea for this problem was to iterate through all of the choices for the coefficients and in , and then for each polynomial plug in values until we hit a composite. Record maximums as appropriate.

There are a few easy optimizations to make. First, itself has to be prime, because we test . Next we test , so we need , that is . Those are the only optimizations I made, though I’m sure there are some more advanced ones out there.

So here’s my code:

# assume PrimeOracle.py is in the same directory as this file
from PrimeOracle import PrimeOracle

def solve(n):
    # seed our oracle to 2*n+2, since n=1 gives 1+a+b
    isprime = PrimeOracle(2*n + 2)
    maxp, maxn = 41, 40
    for b in [m for m in xrange(2, n) if isprime[m] ]:
        for a in xrange(1-b, n):
            idx = 1 # we already tesetd 0 with isprime[b]
            while(isprime[idx*idx+idx*a+b]):
                idx += 1
            if(idx - 1 > maxn):
                maxp, maxn = a*b, idx - 1
    return maxp

I had my oracle also tell me what the biggest number it was asked about was, and the answer was up around 12000. So we could change the starting seed in the oracle to be big enough to never have to extend, and probably save some time.

Suitably to the preceding post I have been pushed to another type of quadratic labyrinth.

This week I received (as a TLS member) the third edition of Labyrinth Pathways. At the back the graphic of this labyrinth was illustrated. I suppose that Jeff Saward has designed it.

A quadratic classical labyrinth

It accentuates the middle. The first path also lies on the middle axis as does the last path into the centre. This is possible by a blank (or enclosed) piece in the ways. The four points of inflexion (= the ends of the boundary lines) form no internal square. Both upper ones are located in the diagonals of the whole square.

Though the well-known basic pattern is totally shifted, however, the labyrinth has 7 circuits and the typical path sequence of the classical labyrinth.

The labyrinth offers many possibilities. Hence, it is amazing and fascinating time and again.

Mi clase de matemáticas es en la madrugada y es pequeño difícil, sobre todo con, más y menos, cincos horas de dormir.

Someone should let me know if that made sense.
Spanish is going great however and should be a breeze as long-archived vocabulary and grammar returns to more active parts of my brain. Mi cabesa trabaja mucho.

En la clase de matemáticas mi cabesa es muy cansada. Quadratics are coming back with a little brute force and some repetition of the base equation so it sticks in my brain. I am too eager in math to jump and try to take shortcuts because of visual relationships that fail to consider orders of operations. This is a habit I need to overcome if Engineering is going to be even a glimmer in the distance. I’m failing math currently.

The crux of my issue is the age old “Why don’t you just measure it with a ruler?” conundrum. Nowhere is this more apparent than in a class full of photographers calculating the area of frames around pictures. Everyone is getting frusterated, which should be comforting (at least it’s not just me), yet I know that the applications of this stuff for my projects is going to be invaluable. When you’re building things like frame jigs (3d renderings later tonight!) you need to compute units and increments in order to build rulers and protractors into the design. I need to know the value of side x of a triangle when I know that one side is 64cm (seat tube) and two of the angles are 73.5 degrees or so.

But why can’t we learn math with real world applications built in from the beginning? Finding ambiguous X makes me want to pull the “Here it is!” stunt. I guess I should just try to work in my math work into my 3d modelling (In Aslar-Vellum Cobalt v8 for the next week and a half!). Let’s start modelling the frame jig on the basis of material surrounding the adjustment slots…hey wait that’s not a bad Idea! In fact that would probably create a jig with minimum waste-strength ratio as well as cut the weight down….Equations and v1 of the jig (sans optimized dimensions for waste reduction) to come later tonight when no estudio por la clase de español.

Hasta luego mis compañeros!

That title is a punctuation FAIL. I’m almost like Margo Roth Spiegelman. Paper Towns, read it.

I thought I should put up a post, because it’s been a while. But really, all I’m doing is avoiding media, maths and music homework. And that isn’t even the ONLY homework I have. Oh, there’s more. Geography. Classics. Actually I can’t think of anything else…oh, Science. But media, maths and music are just the stuff that I plan on doing today.

Can you believe that in WordPress’ Top Blogs, the Celebrity Baby Blog is at Number 4? Who READS that? It can’t be that intertesting! Whatever.

HPOotP was on TV last night (hope you can decipher those letters). I really hate the dumb person that decided it would be an awesome idea to have a silent slow motion scene when Lupin was holding Harry back as he screamed in mourning at Sirius’ death. I also hate JKR for killing Sirius in the first place. Not cool. Actually that movie was all round crappy.

I’ve been trying to remember Japanese lately. 5 hours a fortnight for 2 school years, and I’ve forgotten a lot. I think I still remember all the Hiragana, but as far as actual sentences go. Onamawa nan desuka? Namae wa Alex desu. Tamago. Osutoraria. Then all the others I remember are just words that sound English. Chikin. Caku. Piza. Ich ni san shi go roku. Piza o tabemasu.

I’m bored with you, blog. Goodbye. Ruby if you’re reading this, I know I haven’t replied to your comments yet. In the Look What I Found thing, it was Simone and …Pumba? Was that the ferret thing’s name? I don’t know. I’ve only seen that movie once. Actually…twice. Aladdin is cooler than the Lion King. Genie is cool. Except for the fact that he’s Robin Williams. Have you ever seen RV? That movie is horrible. Horrible. Almost as bad as The Pink Panther. Oh god, that was just…I don’t want to relive it. Those two movies made me hate Robin Williams and Steve Martin forever.

Alright that’s enough. Farewell.

PS: CGU ads are COOL. Wish I was Susan. She’s a providore. She just opened her second store. Try her cheddar, you won’t find better, go on, try some of her Bulgarian Fetta. Whoever you are…Susan! Whatever you do! Providore! We put the U in CGU!

Yeah I know that won’t make sense unless you’ve seen the ads.

Quadratic Factoring

To factor a quadratic expression written in the general form of ax² + bx + c,  means to factor a trinomial into two binomials . The binomials are expressions that when multiplied using the distributive property will result in the trinomial. A trinomial has three terms, an x² term, an x term, and a constant. After factoring, the expression will show the multiplication of two binomials, each with two terms that are added or subtracted.

To break down a trinomial the FOIL method is used in reverse. The steps in the FOIL method are:

First – Multiply the first term in each set of parentheses

Outer – Multiply the outer term in each set of parentheses

Inner – Multiply the inner term in each set of parentheses

Last  – Multiply the last term in each set of parentheses

 Example 1: x² + 5x + 6

 Set up parentheses to show the 2 binomials like this:

( __ + ___ )( ___ + ____)

Start by listing the factors of the first and last term. 5x will be the trickiest one to find so it will be last.

Factors of x² : x

Factors of 6: -6,-3,-2,-1,1,2,3,6

The only way to get x² is to multiply x by itself.

( x__ + ___ )( __x_ + ____)

The next step is to find two factors of 6 that add up to 5 for the term 5x. If the terms in the binomials were negative the middle term would also be negative so we need a positive pair of factors that add up to 5. The only possible factors that fit are 2 and 3.

( x__ + __2_ )( __x_ + _3___)

Now check using the FOIL method.

( x__ + __2_ )( __x_ + _3___)

First: (x)(x)= x²

Outside: (x)(3)=3x

Inside: (2)(x)=2x

Last: (2)(3)=6

x² +3x + 2x + 6= x² +5x+6 This is the original trinomial. All the terms are positive in this example but this is not always the case.

Example 2:

x² -6x +8

Factors of x²: x

Factors of 8: -8,-4,-2,-1,1,2,4,8

The second term is negative so the 2 factors must be negative. Two negative integers added together will equal a negative and multiplied will equal a positive.

( x__ -__2_ )( __x_ -_4___)

Use FOIL to check:

First: (x)(x)= x²

Outside: (x)(-4)=-4x

Inside: (-2)(x)=-2x

Last: (-2)(-4)=8

x² -4x – 2x + 8= x² -6x+8

Example 3:

x² -x -12

Factors of x²: x

Factors of 12: -12,-6,-4,-3,-2,-1,1,2,3,4,6,12 The constant in this example is negative so it must be the product of a negative integers and a positive integer. The 2 factors must add up to -1, the b value.  -4 and 3 are the only factors that add up to -1 while also resulting in -12 when multiplied.

( x__ -__3_ )( __x_ +_4___) 

Did paraphrasing the words help you internalize the concepts more?

Paraphrasing was very helpful because I needed to clearly picture the steps before I could translate them into language my students would understand. I also had to create my own examples and then apply the steps of both factoring and then checking using FOIL to make sure they were accurate examples. That let me think through the process so that I was better prepared to teach this as a lesson.

How can you apply this type of exercise in a lesson for your own students?

Students could create a demonstration for teaching factoring to students in another class using technology such as PowerPoint or iMovie where they can visually show the steps. They could then describe the development process as well as the presentation in their interactive math notebooks or in a journal.

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“There is no better indicator of a species being ready for solar system exploration than the development of the toaster. The toaster demonstrates that a species has developed mechanization, the harnessing of electrical energy for constructive purposes, and the distribution of sliced bread.”
-Professor sqrt((10x²+5x+1)/pi), University of Central Cosine

Universal cycle 4, Subunit 301.405.1: Salutations Compilers, to whom all data is subject and to whom designate all sequential orderings. These are the collected recordings of constructed sentient Y-4514, designation Yort. It is this unit’s distinct pleasure to serve in the eternal struggle of ever recycling observation and analysis. As a servant of the enlightened inquivistivation there is no higher honor than to observe the more highly dynamic civilizations. As such few civilizations observed within this universal cycle are as charmingly schizophrenic and unstable as [humanity].

Pre-infiltration scans of media, and rectal scans of more “dynamic” drones of human society, reveal a fragmented consciousness typical of juvenile species. The most noticeable feature of [homo-sapiens] on a macro level is their tendency to further divide themselves into large, complex units defined as [nations]. [Nations] can be defined as units of societal grouping that are characterized by only a marginal collective hatred between its component units, which is overshadowed by their collective hatred of the component units of other [nations].

Another defining characteristic of nations is their geographical element, in which nations are confined to regions within invisible boundaries or “borders”. It becomes immediately apparent that [homo-spaiens] seem to have a level of hatred and distrust for other [homo-sapiens] that increases in a relative proportionally to the total distance from which they were spawned. [Nations] are thus an abstraction which show the approximate distance of inter-[homo-sapien] cooperation for a given voluntary collective of [homo-spaiens]. [Homo-sapiens] also have the tendency to hate other [homo-spaiens] as a collective group, rather than as individual units, permitting a degree of fluid travel across borders.

I have chosen to infiltrate the most influential of these [nations], in what is arguably the most well-known of their thoroughly average class of hives, that has chosen to refer to itself as [Orlando]. My initial survey of the hive reveals that it’s leadership class is selected by a ritual which involves displaying plumage in the form of placards placed outside dwellings, on transportation arteries, hydrocarbon land/air vehicles, garments, and media transmissions. The [homo-sapien] with the most intricate and distributed plumage is given nearly unrestrained access to the mechanisms of hive development, and are promptly forgotten by the majority of the hive. The process is repeated in a regular cycle, where the current leadership class reminds the hive that they are, to us the [homo-sapien] expression, “heated feces”.

Comparable to the leadership class is what I have decided to refer to as the Edification Erection Arrangers (EEA). They have a symbiotic relationship with the leadership class in which it provides plumage in exchange for nearly exclusive access to communal resources. As mentioned by the initial report fashioned by the the Office of Interstellar Observers, [homo-sapiens] have a unique concept of [private ownership]. In such a relationship, a [homo-sapien] claims exclusive access to a material object which becomes [theirs]. Any unauthorized attempt to access the claimed object is responded to with a varying degree of hostility, in some cases resulting in termination of either the expropriator or the expropriate.

[Homo-sapiens] will reluctantly part with [personal property] to the leadership class for the development of the hive. These resources are then transferred directly to members of the EEA who will develop plans to add on to the hive, and direct drones to do the actual physical work. It seems that what is ultimately constructed under the direction of the EEA is relatively unimportant as long as it is big, costly, and the EEA is allowed to claim ownership of it. Other than the transfer of resources to the EEA the leadership class itself seems largely irrelevant, and seems to be a symbolic position when compared to the power welded by members of the EEA. Further complicating the relationship is the tendency of members of the EEA to also be members of the leadership class as well.

Theoretically members of the leadership class are directly responsible to drones of the hive, which despite widespread disapproval of the leadership class is oddly still the general sentiment of the majority of hive drones. In practice, hive drones are given the opportunity to voice disapproval at minority of the meetings of the leadership class. These sentiments are then acknowledged and openly ignored. From what I can tell, the operation of the management of the hive is very similar to the operation of what [homo-sapiens] refer to as a [business] (a subject which will require much elaboration), in all but the opportunity to be personally ignored by the leadership class.

The general populace of the [Orlando] hive is rather sedentary in most aspects, most notably in its interactions with its leadership class. It is quite a feat that the hive operates at all, as the drones fracture themselves into a myriad of social fragments, with many eschewing social interaction to a degree unfathomable in our [???] society. I am worried that the degree of interaction required to accurately document this one hive may be more than I can spare. The publishing date for the -1st edition of the Omni-pedia of Sentient Species is rapidly approaching, and I dare not face the wrath of the Compilers or miss the next universal cycle. I also fear that infiltration will be delicate and will require transformations between multiple subjects to be able to travel between social fragments without drawing unwanted attention.

To return from that semi-personal tangent (all necessary apologies) back to the subject at hand. Hatred for the leadership class is nearly universal among the drones, although oddly fewer drones despise the members of the EEA and a good number actually revere them. Of the latter, they place undue emphasis on the powers of the hive leaders, and decry something called [socialism], which according to them, is the redistribution of resources by hive leaders to those drones that have a lot of resources to those with relatively few resources. I am uncertain as to how they have come to this seemingly paradoxical conclusion, as the leadership class obviously does the exact opposite. It will require a much higher degree of observation, outside the scope of initial reconnaissance.

As it is not currently a leadership selection cycle, I have taken the precautionary measure of deleting any and all information on the current leadership class from my internal drives. I have also initiated contact with a drone that has been forwarded to me via the Office of Interstellar Observers. It refers to itself as “Branson Fasricci”, and has provided us with subjects that can be used for infiltration. The deal with this particular drone is outside of my Field of Non-Terminal Information which will most certainly result in my undergoing VOLT (Violent Organic Liquification Transformation) if observed. As such, no attempts to observe such information has been attempted.

The first drone that I have been provided with is 3 x 0.15 [non-translatable unit of measurement], 600 [non-translatable unit of measurement], and a somewhat spongy midsection. It is male with a white skin pigment, relative lack of fur, and has an anal cavity of 309 [non-translatable unit of measurement]. Subject is not rated for atmospheric entry, as has been demonstrated with previous models.

As such, per protocol 42 subsection 1.943*10²³, I will be performing atmospheric entry intercourse via quantithermoplastic cylinder. Intercourse will be disguised by standard “Nothing to See Here” protocols. Sperm whales coated in bacon grease are ready for atmospheric deployment at forwarded points alpha, beta, and gamma. In addition to atmospheric nasal overload, we will be deploying big rocks to drop in forward point zeta to distract the drones with pretty streaks of light. Yort out.

Selama ini mengandalkan kalkulator yang kupunyai untuk menyelesaikan persamaan kuadrat (pangkat dua). Eh…pas lagi ada tugas kuliah, ketemu soal yang mengharuskanku menghitung persamaan pangkat empat! Waduh, semasa kuliah S1, apalagi masa SMU, kan ketemunya cuma pangkat dua

Karena ngga familiar dengan hal ini, iseng-iseng browsing di dunia maya untuk mencari kalkulator online yang bisa memecahkan masalahku ini. Alhamdulillah…ketemu juga! Sumber websitenya ada di Calculator City. Rupanya web ini menyediakan banyak fasilitas perhitungan. Ada pengkonversi (panjang, massa, energy, luas, dan lain-lain) dan kalkulator untuk keuangan, geometri, aljabar, trigonometri, kalkulus, astronomi, kimia, fisika, matematika dasar dan lain-lain. Ada juga teori bagaimana menghitung manual di beberapa kalkulator yang disediakan.

Berikut kalkulator yang berhubungan dengan persamaan pangkat dua (quadratic equation), pangkat tiga (cubic equation) dan pangkat empat (quartic equation):

1. Persamaan pangkat dua (Quadratic Equation)

2. Persamaan pangkat tiga (Cubic Equation)

3. Persamaan pangkat empat (Quartic Equation)

Semoga bermanfaat yah!

Quadratic inequalities can often cause problems when trying to decide which range of x values are the solutions. As they are quadratics you usually get two values of x and then have to decide whether it is the values of x between the two bounds that are the solutions or the values of x either side.

For example when solving the inequality:
2x²+x-6>0
we get
(2x-3)(x+2)>0

From this you may be tempted to write x<3/2 and x<-2 however this would be incorrect. The solution must be either -2<3/2, or (x<-2 or x>3/2).
This must be the case for any quadratic with two real roots because all the points between the two roots have one sign and these which are outside the roots have another, as can be clearly seen by considering the general graph.

We can determine which of the two solutions to use by either putting in values to see which work or by drawing a sketch, usually drawing a sketch is the best. For the sketch we only need to consider whether the coefficient of x², in this case 2, is positive or negative so we know which way the graph curves. The y intercept of the graph is irrelevant.

So in our case because 2>=0 the sketch is a “happy” face so the y>0 either side of the roots and therefore our solutions for x are:
x<-2 or x>3/2

By David Woodford

I made a simple worksheet asking students to use graphing calculators to graph quadratic equations in sets of 4. For example, the first set was:

They were then required to sketch the graphs and write a statement about how the graphs were related by transformation.

The final part of the lesson was when I really stretched them. I created a powerpoint with images of graphs I had made using an online graphing calculator and asked students to make their calculator screens match my screen. Without having formalized rules for quadratic equations and graphs my students were able to do it, they discovered the rules and patterns.

Files:

worksheet

powerpoint

In this post, SpaceTW asked:

Hi, I need help aiming at moving targets for a program. I have the variables for the speed and direction of the enemies, but the targeting is still not accurate enough.

There’s a very simple solution if you don’t mind the bullets travelling at different speeds, but that can lead to bullets travelling very fast or very slow, so let’s say that we want every bullet to go the same speed. Here’s the solution:

Suppose you have a tower at a fixed position , and an enemy at initial position moving in the direction .

You want to shoot a bullet travelling at a certain velocity so that it collides with the enemy.

Let the direction of the bullet be .

If and , then the bullet will move in parallel with the enemy, always staying the same distance from it.

You can now consider that the enemy is not moving at all, relative to the bullet, and the difference in position between them is . The bullet can move along this line to collide with the bullet.

If the bullet takes frames to reach the bullet, we now get:

As we said that the bullet’s velocity is , we get

Which expands out to:

(I had to split that over a few lines so it would fit in the post column, sorry!)

Which is a quadratic equation! So set:

Then, using the quadratic formula:

You can now put that back into the equations for and , and you’re done!

Here’s the result, written in BlitzMax:

diffx:double=x2-x1
diffy:double=y2-y1
v:Double=3                'desired speed of bullet

a:Double=diffx*diffx+diffy*diffy
b:Double=2*(vx*diffx+vy*diffy)
c:Double=vx*vx+vy*vy-v*v

tInv#=(-b+Sqr(b*b-4*a*c))/(2*a)   ' solve 1/t
dx=diffx*tInv+vx
dy=diffy*tInv+vy

That’s it!

A few weeks ago, I was writing an email when I caught myself mixing up “their” and “there”. Not only that, but I’d also misspelled two other words and left one sentence incomplete.

Well, this is it, I thought. I’ve finally hit the wall. Burned out. Peaked.

This past week, a college friend and I reminisced about our school days: working hard, playing hard, going out every night, and still finding a way to pass our finals. “We were so smart back then,” she sighed. And so energetic. I went home that evening and crawled into bed by eleven, the new default bedtime for my elderly, depreciating self.

So, yes… two years out of college, at the ripe old age of 23, I am now officially over the hill. In the parabolic chronology of life, I’ve already surpassed the tangent point, the point of no return. Just a few years ago, I was juggling five classes, two extracurriculars, a term-time job, and an active social life. Now, I have one job, no classes, and a schedule interspersed with lengthy midday naps. Gone are the days when I could forgo the spell check and formulate coherent sentences on the first try. I can’t quite remember the quadratic formula and I definitely don’t know how to graph parabolas anymore. All I know is that I’ve passed the peak. Pretty soon I’ll be forgetting my own name and calling people “loosers”.

Sadly, we all must go the way of old geezers a la John Madden. (A gem from last night’s Super Bowl: “If you can become a head coach at any age, you can coach when you’re young.” Thanks, John.) A friend recently sent me an email about “Kitchenheimer’s“, the phenomenon “when you’re in the kitchen going around in circles because you can’t remember what you were doing there.” In the email, he wrote, “I bet this happens to you all the time.”

It does. In my defense, I’ll pull an anti-Benjamin Button and blame it on accelerated aging. But for everyone else, I hope you fare better than me. Here’s a roadmap to what the future holds from a work perspective, as you traverse across the x-axis through the parabola of your long and storied career.

.

(1) Hustlin’: Young, eager, energetic, and bright… full of vitality, ambition, and grandiose ideas.

You are still in this phase if you can do any of the following:

• Make it through a week of work without drinking coffee
• Get really excited about networking, company dinners, and conversations about “synergy”
• Need less than 6 hours of sleep to function
• Start a foundation to save polar bears in your free time

(2) Peaking: Slowing down, spacing out, and taking more bathroom breaks… The only hope is to hover around the top before the rapid freefall.

You may be peaking if you do any of the following:

• Consider work with a combination of bemusement and utter apathy
• Spend at least 15% of your time at work on fantasy football/eBay/Perez Hilton
• Take more than 20 seconds to remember the work you did yesterday

(3) All Downhill From Here: Lethargy, complacency, and embracing monotony… Work has become as exciting as pressing a button every 108 minutes. Boop.

You may be on your way to sloth-ville if you do any of the following:

• Volunteer to be laid off even though no one asked you
• Think daily about pulling a Milton from Office Space
• Derive your greatest joy by stealing something from the office every day

Now, this same graph can be applied to other aspects of life as well: for example, that loving feeling. Just like productivity, acuity, and energy, our loving aptitude should decline as we age. Then again, with technology these days, a trip to the pharmacy can extend the hustler stage and keep you at your peak for much longer… (pun intended).

So, will there ever be such a development for our work lives as well? Can I come back from futility and rid myself of spell check dependence once again? Will the Type-A ambition return? Will quadratics be superseded by cubics? (If you get the last reference, then be assured that you are definitely still hustlin’.)

I’m hoping yes. Forget about Kitchenheimer’s… If John Madden can make it work, then I should at least be able to save some polar bears before it all goes downhill for good.

If you want an explanation on using the formula go to the quadratic formula post (with a downloadable solver)

The quadratic formula,

for the general equation ax²+bx+c=0

can be derived using the method of “completing the square” as follows.
starting with

ax²+bx+c=0

divide through by a

x²+bx/a+c/a=0

take the c part to the other side

x²+bx/a=-c/a

In order to complete the square we need to add half the coefficient of x (that’s b/2a) squared to both sides so we get

x²+bx/a + b²/4a²=-c/a +b²/4a²

Now we can factorise the left into a squared bracket (you can check this by multiplying it back out)

(x + b/2a)²=-c/a +b²/4a²

So if we square root both sides and take b/2a to the other side we have x on its own

x = -b/2a ±√(-c/a +b²/4a²)

Now to get this in the more traditional form we can take out 1/2a (Which means each term in the root is timesed by 4a²) from the square root to put the whole thing over 2a

x = (-b ±√(b²-4ac))/2a

David Woodford

It has been nearly a week since I uploaded it to the App Store until it’s available. But it’s ready for sell finally!

SciCal Pro 1.0

This is something you can do with the Full version:

Solving quadratic equation.

- Solving quadratic equation:

All you have to do is enter the three coefficient a,b and c and you it will solve any quadratic equation you can think of. You can also use it to factor any quadratic expression. Just use this formula here: ax^2 + bx + c = (x – x1)(x – x2). (x1 and x2 is 2 solutions of the equation)

Solving system of linear equations.

Solving system of linear equations.

With this feature you can also find the intercept point of any 2 lines in an 2d plane.

I’m trying to add another feature which can solve system of equations of three variables.

a1*x+b1*y+c1*z = d1

a2*x+b2*y+c2*z = d2

a3*x+b3*y+c3*z = d3

It will be available soon!

Integral Screenshot

Yes! It can find the value of any kind of definite integral! (Caution: this is the approximate value!)

Beware that the more complicated the expression is, the more time it takes. Sometime the application may crash if the device has low memory. All you have to do is restart the application. (or the iPhone/iPod Touch in worst case)

But this is useful enough to check the answer of your integration!

In the near future it can do something like this:

Example

I’m working on it. (The application in the example is not mine)

Please let me know if the application has any this part that disappoint you

In this post the procedures described for C++ here will be replicated for Fortran:

The spreadsheet, fortran source code, and compiled dll described in this code may be downloaded from Cubic.zip

As always, it is better to use the downloaded code, rather than copying from the post, because the WordPress software changes quotes (“”) and several other characters, so that code copied and pasted from the post will often not run without editing.

This post describes:

The Fortran code for the quadratic function (QUADA) is shown below:


F_stdcall FUNCTION QUADA(qdat, ResA, NumRows)
IMPLICIT NONE

INTEGER, intent(in) :: NumRows
REAL(2), intent(in) :: qdat(NumRows, 3)
REAL(2), intent(out) :: ResA(NumRows,2)

REAL(2) :: QUADA
REAL(2) :: P, DIS

INTEGER I

DO I=1, NumRows

IF (qdat(I,1).NE.0.D+0) THEN

! quadratic problem
P=5.D-1*qdat(I,2)/qdat(I,1)
DIS=P**2-qdat(I,3)/qdat(I,1)
IF (DIS.GE.0.D+0) THEN
! two real solutions!
ResA(I,1)=-P-SQRT(DIS)
Resa(I,2)=-P+SQRT(DIS)
QUADA=2
ELSE
! no real solution!
QUADA=0
END IF

ELSE IF (qdat(I,2).NE.0.D+0) THEN

! linear equation
ResA(I,1)=-qdat(I,3)/qdat(I,2)
QUADA=1

ELSE
! no equation
QUADA=0
END IF
END DO

RETURN
END FUNCTION QUADA

Note that:

• The function is preceded with “F_stdcall”
• The function parameters: qdat, ResA, and NumRows are declared as either intent(in) or intent(out); intent(inout) is also allowed.
• The arrays qdat and ResA are dynamic, and will be sized at run time to the size specified in NumRows
• Paramaters that are specified as double in VBA are REAL(2) in Fortran
• Parameters specified as Long in VBA are Integers in Fortran
• It is essential to ensure that both the data types and the sizes of arrays in the VBA and Fortran routines match.

The fortran routine is compiled as a dll as before, and may then be called from a VBA routine.  The dll has been named cubic.dll, since it contains both the quadratic and cubic functions:


Declare Function QUADA Lib "D:\Users\...\Cubic\Release\Win32\Cubic.dll" (qdat As Double, ResA As Double, NumRows As Long) As Double

Function FQuada(QuadData As Variant) As Variant
Dim xa() As Double
Dim A As Double, b As Double, c As Double
Dim ResA() As Double, NumRows As Long
Dim Retn As Double, i As Long, j As Long
On Error Resume Next

QuadData = QuadData.Value2
NumRows = UBound(QuadData) – LBound(QuadData) + 1

ReDim xa(1 To NumRows, 1 To 3)
ReDim ResA(1 To NumRows, 1 To 2)
For i = 1 To NumRows
For j = 1 To 3
xa(i, j) = QuadData(i, j)
Next j
Next i
Retn = QUADA(xa(1, 1), ResA(1, 1), NumRows)

FQuada = ResA

End Function

As for C++ dlls, array parameters are passed by reference by entering the first member of the array. Unlike C++, in Fortran the array members are named in the same way as VBA; i.e. ArrayName(Row No, Column No).

A screen shot showing the output of the quadratic function is shown below:

Output from QuadA

Fortran code for two Cubic functions, and the resulting dll, are included in the download files.  The code to call these functions from VBA is shown below:
Declare Function CUBIC Lib “D:\Users\…\Cubic\Release\Win32\Cubic.dll” (cdat As Double, ResA As Double) As Double
Declare Function CUBICA Lib “D:\Users\..,\Cubic\Release\Win32\Cubic.dll” (cdat As Double, ResA As Double, NumRows As Long) As Double

Function FCUBIC(P As Variant) As Variant
Dim P1(1 To 4) As Double, Res(1 To 1, 1 To 3) As Double
Dim cval As Double, i As Long

P = P.Value

For i = 1 To 4
P1(i) = P(1, i)
Next i

cval = CUBIC(P1(1), Res(1, 1))

FCUBIC = Res

End Function

Function FCubica(CubicData As Variant) As Variant
Dim xa(1 To 4) As Double
Dim ResA1() As Double, ResA(1 To 1, 1 To 3) As Double, NumRows As Long
Dim Retn As Double, i As Long, j As Long
On Error Resume Next

CubicData = CubicData.Value2
NumRows = UBound(CubicData) – LBound(CubicData) + 1

ReDim ResA1(1 To NumRows, 1 To 3)
For i = 1 To NumRows
For j = 1 To 4
xa(j) = CubicData(i, j)
Next j
Retn = CUBIC(xa(1), ResA(1, 1))
For j = 1 To 3
ResA1(i, j) = ResA(1, j)
Next j
Next i

FCubica = ResA1

End Function

Function FCubica2(CubicData As Variant) As Variant
Dim xa() As Double
Dim ResA() As Double, NumRows As Long
Dim Retn As Long, i As Long, j As Long
On Error Resume Next

CubicData = CubicData.Value2
NumRows = UBound(CubicData) – LBound(CubicData) + 1
ReDim xa(1 To NumRows, 1 To 4)
ReDim ResA(1 To NumRows, 1 To 3)
For i = 1 To NumRows
For j = 1 To 4
xa(i, j) = CubicData(i, j)
Next j
Next i
Retn = CUBICA(xa(1, 1), ResA(1, 1), NumRows)

FCubica2 = ResA

End Function

The code includes one function that operates on one equation at a time, and two alternative array functions that provide much better performance by avoiding the transfer of data between the spreadsheet and VBA. For comparison VBA cubic functions are also included in the download file.

Screenshot of cubic function output and relative performance results:

Following up yesterday’s post, I’ve spent some time with factoring quartics as the composition of two quadratics, and looking at the quadratics in vertex form to get a better graphical understanding of the problem. Also, yesterday I mentioned the apparent dis-symmetry in the freedom allowed when choosing as one of the free parameters, versus choosing . I realized that came from the assumption that the leading coefficient for the quartic was positive, and probably has no further bearing on the problem.

So, let’s get back to it, as simple as possible. Our overall goal is to write as the composition where and are quadratics. Since we now know we have enough freedom in choosing some of the coefficients, without losing any power, let’s go ahead and assume that both and are monic. I think quadratics are easier to visualize in vertex form, so write , (so the vertex for is at ).

Since we’re also allowed to freely choose the constant term for , let’s choose it to be 0. This has several wonderful implications. The constant term for is , so if this is 0, we can set to eliminate a parameter. Now that the constant term for is 0, and since we’re assuming the constant term for the composite is also 0, we see that the constant term for is also 0. Thus, , and we have written . This means the vertices for the two parabolas lie on the parabola , which is somehow visually appealing. We can picture two parabolas opening up with vertices travelling around on , and make a nice movie of corresponding quartic compositions. Perhaps I’ll see what I can rig up in Maple, or something.

With these formulas for and , some things are quickly apparent. It’s comforting to check that the relation (from yesterday) is satisfied by the coefficients of the composite (here, are the coefficients of in the composite). Next, notice that

which is an even function (symmetric about the -axis). Since is just a horizontal shift of (shifting left units), this means that is symmetric around . Unless I’ve got some implications wrong somewhere, I’m pretty sure what we’ve just shown is that a quartic can be written as the composite of two quadratics precisely when it is symmetric around a vertical axis. But I’d like to double check that statement (or let you).

Suppose you had a fixed quartic (with ) and wanted to identify the and for the quadratics that we’re talking about. It’s easy to do, with the work from yesterday. Yesterday, we said that the linear coefficient of was equal to . Thus, , so . We also found that the linear coefficient of would be . So , so , which is if . Anyway, the point is that the vertices for the quadratics can be easily picked out from the coefficients of the quartic you started with.

Finally, a nice thing about the constant terms being 0 for these two quadratics is that then , so one of the roots is 0, and the other is . So the composite, will be 0 when (so ) or when (so ). It’s also not hard to find the zeros for the first and second derivatives for . The first derivative is 0 at and , and the second derivative is 0 at . Of course, not all of these values in square roots need be positive, but whatever. The symmetry around is readily apparent.

So, I’m feeling good about factoring quartics as quadratics. I’ve got no real questions remaining, personally. If you do, please let me know. What’s next? Well, there’s no point trying to factor a quintic as a composite, because 5 is prime, so the next logical choice would be a sixth degree polynomial as a quadratic and a cubic. As fun as that sounds, I may leave it to somebody else. I’ve got another project in mind in relation to polynomials, but you’ll just have to come back and find out about that later.

I recently wrote up a piece about factoring a given quartic polynomial as the composition of quadratics on my blogger page. I’ve been playing with it some more, and can now say things slightly better. On that page, though, I noted that we can assume that the quartic we start with is monic, with constant term 0, so I continue to assume that. I won’t go through much of the ‘proof’ of what follows, at it is covered on that blogger page.

Prop: A quartic polynomial can be factored as the composition of two quadratics iff . Moreover, when this relation on the coefficients is satisfied, there are a two-parameter family of solutions for a factorization.

Suppose the relation is satisfied, and define

The notation comes from assuming , . In the above, we have let and be free parameters. One could, alternatively, use and as the free parameters, obtaining polynomials

Clearly this requires . It’s easy enough to check that the two pairs of polynomials compose to give the original quartic.

It seems like there should be something to say about the fact that the free parameter is more restricted (positive) than if we had chosed as the free parameter (it just has to be non-zero). Sadly, I don’t know what should be said.

I’ll probably continue playing with this a little. I like thinking about quadratics coming from vertex form, , as opposed to the standard form, used above. It’s easier to see where the quadratic is, for me, in vertex form. So I’d like to get a better visual understanding of the factorization, if I can. I smell simplifying assumption and Maple animations.

在下每年幾乎都要教四個課程，同工一定了解，到時到侯就是測驗「旺季」，同時段出三四份卷，批改五六份卷，常有之；案桌堆積的功課作業，是欠學生的迷債；如果授課員是有家室之人（像我），時間很可能是不由自主（以致工作計劃告吹又告吹），只能嘆一聲「人在江湖」。

入正題，以下是改卷時看到的。

這是一道中四數學題：

If the graph of touches the graph of at one point, evaluate .

答案是 。技巧不過是代入法或消去法云云，再考慮判別式等於零諸如此類。

但部分同學給出以下做法：

把 寫成 subject，從而消去 ，即

– - – - – - (*)

之後，他們用

，得

代入 (*)，得 ，從而得出

這個錯誤答案。

習題
試評論上述做法之不妥之處。

其實要感謝學生，他們讓授課員知道還有教什麼。