Q1: finding the roots

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A straight line is expressed as y = a x + b which is  a first degree equation since the power of x is 1.

Now what if the power is 2.

​That is precisely what you see in this diagram where y is equal to the second degree of x. 


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If you simply multiply two linear equations you end up with a second degree equation as shown here. 

A little discussion on the words EQUATION, EXPRESSION, FUNCTION is in order here and must be understood. An expression is simply an algebraic entity e.g. x+1.

Now we want to give it a name, so we call it by a single alphabet name, y. And that is how we get this:     y = x+1.   As soon as we do that, a graphical world opens out in front of us which has the y axis and the x axis. Always think of y as the VERTICAL direction and x as the HORIZONTAL direction. Because that is what they are. 

For each value of x as you go right from 0 to 1, 2, 15, 59 … to infinity,
each time correspondingly, there is a y value given by that expression y=x+1. And you plot these on the graph paper and arrive at, in this case, the straight line.
Since y, at each point of x depends on this expression, we say y IS A FUNCTION OF x.
This is a very important way of reckoning this thing. Once you get this, we will henceforth use this word, function, most of the time: y is a function of x.

Another way of understanding this phrase is that y depends on x. It is like saying your grade is a function of your understanding. 

And we write it as follows: y = f (x) = x+1, in this case. 

So, whenever you see or hear of this: f (x), you should be reminded of two things: an algebraic expression and 2) a graph where f (x) is actually the y values of the expression and gives you a beautiful diagram. 

So the basic quadratic function that we see is a U shaped diagram. 

Next we look into the different parts of the diagram, the curve, the function, the figure: different names, same meaning. 

Particularly, we look at the points where the curve cuts the x axis and the y axis. We need to do some algebra for that. Read on…

[Don’t worry if you do not get the concept of function completely. It will become clear as we go along: for now remember, function means the graph and the algebraic ]

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Here we show how a quadratic expression is actually a product of three elements: a, x minus alpha and x minus beta. 


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You have seen above that two linear expressions when multiplied give rise to the quadratic expression containing the second degree of x i.e. x to the power of 2. 

Here the reverse has been engineered. A general quadratic expression has been taken and we separate it into the product of two linear expressions. Some algebraic jugglery was involved as you can see.


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Continuing from the previous diagram, we arrive at the two linear expressions and finally obtain

a(A+B)(A-B) where 
A=x + (b/2a) and
B=(D/2a) where
D is the square root of the expression b^2 – 4ac.

We will have much more to say about D in subsequent lessons. For now reckon the simple form A+B and A-B. 

An important skill is to look at the complex as combinations of simple structures. If you can identify A+B & A-B, no matter what A and B are, even if they are complex, you know that they break out into these simple forms. Maths is much to do with FORMS. That is why we use the word FORMULA so often in Mathematics.