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May 27th, 2008
February 27th, 2008
http://www.geocities.com/dong_zemao/Quad
February 11th, 2008
They are used to represent roots of a quadratic equation. if (x-5)(x-4)=0, then the roots are 5 & 4. (But you already knew that, don't you?)
Sometimes roots are difficult to find. Sometimes they are complex. But their sum and product is still real! Amazing? Not so. They are just some properties of a quadratic equation.
Let this general equation of roots A & B (alpha and beta hard to compute here...) be y = ax² + bx + c.
The sum of roots, A+B = -b/a.
The product of roots, AB = c/a
Basically everything else is a combo of the 2.
A²+B² =(A+B)² - 2AB
A²B+B²A = (A+B)AB
etc...
Bonus Question: This is from your textbook. (A^4)+(B^4) = ?
Remember A^4 means A to the power of 4.
What about Questions that asks you to form new equations with new roots? like roots 4A, 4B?
Well, you must first remember that all quadratic equations have the properties mentioned above.
So let this equation be x²+hx+k = 0
Sum of roots 4A+4B = -h/1
Product of roots (4A)(4B)=k/1
Substitute the values in again to get h & k...then you'll get the new equation!
That is the crux of this part. Difficult at first, however this will soon become a breeze.
February 1st, 2008
January 31st, 2008
Sunny days
Tis' what I longed for
Under trees. Pathways
Dancing. Keeping score.
Yesterdays and forever more.
Flitting bees
Orange flowers
Rolling breeze and book towers.
Morning glories
Youth and warmth
Trails of soil
Ever in my path to
Secret Garden where they stay
Twisted minds, and moody grey
January 29th, 2008
Mammoth load it weighs
Asunder, those people torn
Through the last of winter days
Have the teachers been struck with craze?
'Till the end of the time', they say
Even 'furthermore' today
Sudden stress upon their heads
Trudging on towards their beds
Must the tragic be repeated,
On another time we choose?
None deserves this suffering,
Due to what they lose.
Acrostic poems are so much fun
You should try it too!
(Clever friends, Read Vertically.)
What to Study for : Chapter 1 - Simultaneous Eqns, Remainder Theorem, Factor Theorem
January 23rd, 2008
For example, when x² + 2x +2 is divided by x + 1, the remainder R is (-1)² + 2(-1)+ 2 = 1.
Moreover, the division process can be written in reverse into a multiplication identity:
f(x) = (ax -b)Q(x) + R
where Q(x) is the quotient.
so if x = b/a, (ax-b) = 0, so f(x) = R. Hence the Remainder Theorem.
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I could still recall the Great Deportation of 2004. A certain institution deported many of its students for achieving below the minimum required results. Some thought that this was cruel; some agree that this is the only way to ensure our institution's achievements.
Others asked me for opinions. I recalled saying, " It is not the results that is the most important, but the attitude." Though most agree that the deported ones seem to lack both, there are some who could use some salvation.
Years passed, yet the situation remained. I tirelessly tried to feel compassion for people, trying to understand their actions. My compassion has repeatedly been interpreted as leniency, and this leniency misinterpreted as a weakness. I feel a deep sense of remorse.
I'll declare this here and now: When the time comes, the docile writer will draw his weapon, and slay all who stands in his way. With reason on his left, and faith on his right, he will stand victorious.
Thus spoke Lowbeam.
humiliatedJanuary 21st, 2008
Looks simple. The only thing that happens in identities is to compare an unknown term, like Ax², and a known term, like 5x². Since it works for all x values, A must be 5. Its that simple!
Rather, it is the identity of myself that I fear of losing. I am becoming more unlike myself in this journey of lectures. Am I evolving? Or am I dissolving into this rancid pool of conformity? I really cannot differentiate, much less integrate into this seemingly homogeneous society.
Eccentricity has been my forte. I hope it'll continue to serve me well.
January 20th, 2008
Anyway, I've decided to slow down the pace by elaborating more on division of polynomials. It is difficult for learners because of the constant confusion during long division. Just recall long division with numbers and it'll be much simpler. More practice and attention is all needed.
Saturdays are either 'Stay-Home-And-Rot' days, or 'Cafe-Chim-Talk' days. Today (19-01-08) was one of the CCT days. The four of us met in our usual meeting place, XYZ cafe, after dinner. The main discussion today was about the network theories in a book one of my friends read. This was the research that brought about the famous 'Six Degree Of Seperation' theory, in which everyone is linked to everyone else in the world by a maximum of 6 'jumps'.
Illustration: Me - Person1 - Person 2 - Person 3 - Person 4 - Person 5 - You
Of course we all have heard of this, but research on it? I thought it was a for fun thing, like friendster or something. But I was wrong.
It seems that there are lots of uses for it. Finding the general links between items of a large group, like people, can determine how information / diseases / other stuff is spread. Who is the Linking Hub? Who's the Ulu one? By locating the crucial points in this network, one can efficiently know the precise points to minimise the damages. Moreoever it can be used for many other forms of networks.
The other of us could not accept such an arbituarily derived method, of course, and a 3-hr long debate came to discuss the usefulness and practicality. It was difficult to be convinced that such a thing is a root to the majority of our statistical problems. Till now, there has been no clear stand established. I presume this may be carried forward to next week.
Looking forward.
January 14th, 2008
Rushing. Marking. Running. Meeting.
These are what engulfs my staff room now. Who says we are free after class. There are many things to be done. Typing and Marking, and picking out common mistakes for notes. All at once. That is what I call simultaneity.
Soon, the 'more thans' and the 'less thans' will be over. It is time we embark on a new journey with A Maths. The land of SiMUltaneous equations, Remainder theorem and Factor theorem (SMURF*).
Firstly, simultaneous equations will be solved. These involve more than 1 variable. Generally, the same number of equations as there are to variables is required to solve completely. (3 variables need 3 equations at least.) It can be both linear ones, or non-linear.
Secondly, algebraic EXPRESSIONS and POLYNOMIALS will be taught. They should be arranged in a specific descending / ascending order, like x² +2x +1 or 1+ 2x + x². x² terms should always add /subtract x² terms. Division of polynomials by other polynomials should always end in the form F(x) / P(x) = Q(x) with remainder R(x).
Also, F(x) = P(x)Q(x) + R(x). Note that the remainder R(x) should always be at least one degree less than P(x), the divisor. (If P(x) is x², R(x) can be bx, or c, but not ax²)
Thirdly, identities will be taught. Recall things like first law of algebra. This is an example of an identity.
Fourthly, Remainder Theorem. It states that when F(x) is divided by a linear divisor ax - b (a not 0), then the remainder is F(b/a). Together with this is the factor theorem, which just states that if F(b/a) = 0, (ax - b) is a factor, and vice versa.
Lastly, factorising cubic expressions. With all that we've learnt before, it is now time to use them.
Lots to teach. So little time! This will really be a tough topic.
*This abbreviation is entirely copyrighted to Prof. Lowbeam Xavier. Any plagiarism will be seriously dealt with.**
**Copyright Limited to Particular School Only. :D
January 11th, 2008
The basic stuff of any shape. Of course lines come from dots, but they're still quite basic.
However, many people do not know how to draw lines.
Whether it is the curved lines of quadratic equations. Or the number lines of inequalities.
Where it starts; where it ends...
So, I had to draw the line for them.
Where they can start; where they SHALL END
I always lament, when I have to exert authority. I am beginning to become what I dislike : an authority.
Yet the line shall now be drawn.
On a lighter note, I am already finishing linear inequalities soon. Its fairly simple, and the combined topic test will be on the fourrth week. Crossing my fingers now, because I really hope no one will need remedial lesson.
A lazy teacher's wish. :|
lazyJanuary 10th, 2008
well most of it is really what people know in daily lives, in symbols and signs. Lke if x > z, and z>y, then x > y.
Then there is addition / subtraction. if x >y, then x+a>y+a and x-a> y-a, 'a' can be positive, negative or zero.
There is also the multiplication / division. However, whether the extra term is positive or not matters.
e.g. if x>y,
ax > xy, x/a > y/a if a>0,
ax < xy, x/a < y/a if a<0
Also, there is the addition of 2 inequalities.
e.g. if x > y, a > b
then x+a > y+b
However, if x+a > y+b
it does not mean x > y and a > b
The relation is in 1 direction only.
And, if x > y, a > b
It does not mean x -a > y - b
January 9th, 2008
Finally, we reach the part that intrigues students for ages. What is the use of learning all these?
Typically there are 2 kinds.
1. Those that require 2 unknowns related in a known way (e.g. x and x-2) and you know their product
(e.g. x(x-2) = 4)
This is the easy one.
2. Those that require 2 unknowns related in a known way, and you know the product of their reciprocals.
(e.g. 5/x +3/(x-2) = 2)
This requires the change from fractional equations to quadratic equations.
Either way the method had been taught. Only the identification of the equation is needed.
P.S. My problem returns. I'm caring too much about small details. It is difficult to shed the habit, because as a scientist, I have to be careful with any observations. Seems that I don't have the talents to be a teacher, especially an indifferent one.
January 8th, 2008
But it is also really exciting.
More or less like bonsai growing, but much faster. And I get paid to do this! lol
Of course the pay is not all that matters. It is to see the faces of students who struggle through the thick darkness, only to see the light and emerge into civilisation.
Yet the messenger tires himself. Reduced. For the sake of a greater future, he must work hard. To pay for future debts.
Not only is the messenger reducible. Equations are also reducible. Fractional equations.
1/x² +2/x = -1
How does one solve this?
Well you can choose to substitute or multiply through out by x²
1/x² +2/x = -1
Substitute:
let y = 1/x => y² = 1/x²
=> y² +2y = -1
y² +2y+1=0
y= -1
x=1/y
=1/-1
=-1
Multiply by x²
1+2x=-x²
x²+2x+1=0
x=-1
Quite Easily Done.
Only problem? These methods won't work if you get y=0, or x=0.
In either case, 1/y (which is x), and 1/x would be undefined. Then they can't be real.
So if such answers occur, you can reject them! Otherwise, General Solution all the way!
amusedJanuary 7th, 2008
I still remember my days, where this formula is remembered by doing lots of questions. But, I should not do this to them. They seem very intelligent, and are capable of comprehending me to a large extent. If only they would listen.
Its the first week and the problem is recurring again. What is left from NS is a scar and a pressure. Well, they are trying it too. Maybe its what humans are. To hurt the wounded in the wound. If only they would listen.
P.S. b²- 4ac is called the discriminant.
depressedYeah. Thats me. Me with my checkered shirts and black pants. This is totally ruining my reputation as a young thinker!
Well, to couple with it, I had to teach 'Completing the Squares' the other day. Irony? Definitely.
They're learning the method, which is much simpler and more redundant to use, because the questions comprises of an x^2 term, and an x term. Something like x^2 + kx.
It is rather obvious that factorising is no kick, but I had to make them do the longer method. Well, its training, and the more advanced questions may require them to be proficient with the method.
To complete the square, we must recall the identity : (x+p)^2 = x^2 +2px + p^2. They call it the first law of algebra. I... seem to already forget when I learnt this.
So, to solve for x in this method, u must factorise the terms into a square ( x + something )^2. From the identity, we can see that whatever u want to put for the 'something', the coefficient of x is double of that 'something'. So, if the coefficient is k, then the 'something' is 0.5k. (I prefer k/2, but fraction in this blog is confusing, so I use 0.5k for now.)
We also see that we're missing the last term with no x on it. It is the square of the 'something'. But we cannot add in the term. It'll make the whole thing unbalanced, like saying 0 = 0 + (extra term).
Thus, we take the same term (square of the 'something') away:
x^2 +kx = x^2 +kx + (0.5k)^2 - (0.5k)^2
Why must we do all this, since we are subtracting what we add? So that we can factorise the first part:
x^2 +kx + (0.5k)^2 - (0.5k)^2 = (x+ 0.5k)^2 - (0.5k)^2
Looks much neater for x, isn't it? Only 1 square term, making it easier to work with. Lets do this with real numbers:
x^2 +8x = x^2 +8x + 4^2 - 4^2, since 4 is half of 8
= (x + 4)^2 - 16 <-----See? Done!
So...what now? Well, depending on the question, x^2 + kx can be equated to any number.
If x^2 +kx + 4 = 0,
Then x^2 + kx = -4
Complete the Square now...
(x+0.5k)^2 - (0.5k)^2 = -4
Pull the non-x terms over to the RHS:
(x+0.5k)^2 = (0.5k)^2 - 4
= 0.25(k^2) - 4
Square root both sides and pull the 0.5k on the left to the right. (SQRT is square root sign)
(x+ 0.5k) = SQRT[0.25(k^2) - 4] or - SQRT[0.25(k^2) - 4]
x = -0.5k + SQRT[0.25(k^2) - 4] or -0.5k - SQRT[0.25(k^2) - 4]
Presto! x is solved! Now have fun with all the practices!
Coming Up Next: The General Solution to Quadratic Equations.
After that, they'll want to destroy me for not teaching them this easy way first. We'll see which method is faster...
January 4th, 2008
I have 3 classes now. D, F and J. D are dominant with girls. The opposite occurs in J. F, interestingly, is right in the middle...well, almost. And you can see how this seems like a strange correlation with their response to my presence. D laughs typically. J proactively speaks. F...Not so sure yet. Only had a lesson with them. Will monitor them closely. Thankfully, according to Mdm T
The first lesson I taught them was not Quadratic Equations, but the fact that asking questions is everyone's right. I being Socrates' fan will not be happy with just passive learners. I'll try to cultivate the passion for knowledge in them. While many view this impossible, that is where the existentialist despair and faith come in. To do something so absurdly viewed by others but stubbornly by me is faith, in both the students and myself.
Second comes the revision in Quadratic Equations. I revised with them the evergreen method of 'smiling and frowning' - If the coefficient of x^2 is positive, then it smiles :). If the coefficient is negative, it frowns :(.
Also, there is this 'Zero Product' Rule that nobody knows of by its name. However, when I told them what it meant, the 'Oh...', 'Yeah, that huh...' appeared. It simply refers to the fact that if P X Q= 0, then P= 0 or Q = 0. Some of them like to add 'Either' into their answers, but it is totally unnecessary. 'Or' will do.
The only new thing I taught them (I think) is roots of an equation. Roots are actually just the points where the quadratic equation y = ax^2 + bx + c cuts the x- axis, which is where y is 0. Equating ax^2 + bx +c to 0, the solutions(x = something or x = something else) will be the roots of the equation. Alternatively, with the roots, lets say, k and m, the equation can be found in the following manner:
since k and m are roots of the equation,
x = k or x = m.
x - k = 0 or x - m = 0
(x - k)(x - m)=0
... and you expand the terms in the brackets to get the answer required.
That is about all that I've taught for the first lesson. I also gave them my contact methods. Hopefully my phone rings.