Q2: The Discriminant, D


We continue from the previous lessons and summarize what we had achieved therein when we engineered a standard quadratic expression to break down into two linear expressions A+B and A – B
where B contains a guy called D, in short and in long form is called DISCRIMINANT. This is an important guy and we need to get introduced to him. 


This is another way of looking at the quadratic expression. When we equate it to zero, we can arrive at the two values of x as shown. The +- sign comes in front of a root, as you know, since the square of both -ve & +ve numbers become +ve after squaring. 

Equating an expression to zero means that y=0 since y is the name of the expression i.e. y is f(x), y is a function of x and that function is the expression, in this case, the quadratic expression. So, when the expression = 0, what it means is that y=0 and if you remember, what is represented in coordinate geometry as y=0? Yes you are right if you said it is the X AXIS. The technical name of the x-axis is y=0 because anywhere on the x-axis, the value of x keeps changing but the value of y always remains ZERO.
Similarly remember that the technical name of y-axis is x=0. 

So, finally, when we put an expression equal to zero i.e. y=0, graphically what happens is we find those values of x where y=0

And that is precisely those points where the graph cuts the x axis. 

This understanding is very very important:
​that to put any algebraic expression equal to 0 precisely means where the graph of that function cuts the x-axis. 

The role of D: DISCRIMINANT: see where it cuts the x-axis


We will deal with the formula shown and D is the guy we need to get familiar with as it influences the geometry. The U shaped diagram of the quadratic can cut the x axis at two places, can touch it at one point or be above the x axis. This is shown in this diagram. Now we can see in the equation that we have two values of x and inside the formula there is a guy called square root of D. And we have identified three cases.
When D is zero, x has only one value i.e. – b/2a. Since x has only one value, it is the diagram II where the U touches x axis at only one point

When D is positive i.e. 4, 40, 54, whatever, it will have a root and it will have two roots +& -, as you know. So, x will have two values and it is our third diagram where the U cuts the x axis at two points. 

And what if D is less than zero i.e. -1, -33 -2.65 or whatever -ve value. What happens then. Remember that a negative number has no Real roots since the square of both + ve and – ve numbers become positive upon squaring. Thus there are no values of square root of D and thus when D is negative, x does not have a Real value and thus, it will not cut the x axis. 

Vital Math Skill: Algebra <-> Geometry
Look at the diagram once again. Now look at the algebra. The skill that you must develop is to see how the diagram comes from the algebra and also how the algebra comes from the diagram. You may think I am telling you the same thing twice by inverting the sentence. But, we will show you both by experimenting with the elements in the graph as well as by experimenting with the elements in the algebra and see how a tweak in algebra affects the geometry and vice versa. Once we can do both of these things, we will have a 360 degree view of any concept in mathematics. In fact, this is the strength which distinguishes mathematicians from others.