Q3: a + -ve, alpha, beta

We have ​dealt with this diagram in the previous lesson and saw that the D affects the placement of the graph of the quadratic function. Precisely, the value of D tells us whether the graph will cut the x axis or not and also if it cuts, whether it will cut the x axis at one point or two points.

​To generalize, the value of D tells us whether the graph of the quadratic function will cut the x axis at 0, 1 or 2 points depending upon whether D is – ve , zero or + ve. Simple, isn’t it ?

﻿In the above graph we had assumed ‘a’ as positive. The fact of the matter is that if a is negative, the graph gets inverted. You can check it out by taking an example on your own with negative ‘a’.

And the entire idea of D telling us the where the graph will cut the x axis remains the same. D<0 implies the graph cuts at 0 points
D=0 implies the graph cuts at 1 point &
D>0 implies the graph cuts at 2 points on the x-axis.

​The diagram makes it clear.

The two values of x: Alpha and Beta

We know that the solution of the quadratic equation gives us two values of x which is of the nature of x= A + – B i.e. one value is A+ B while the other value is A – B. and these are called alpha & beta. What is important to note is the graphical understanding of the two parts of alpha & beta viz. A & B. The A is the mid point of the two roots i.e. x=-b/2a and from that point go right on the x axis by B to get A+B=beta and go left from x=A or the point (A, 0) leftwards upto (A – B, 0) where A – B is called alpha. See it all graphically in the diagram here and internalize.