Ok, so you should know that the discriminant determines how many solutions there are going to be.

For no solutions: .

For one solution: .

For two solutions: .

Let's start with where

.

So for one solution, or .

It's a little harder working out the values of for which there are no solutions or two solutions.

Have you heard of the absolute value function?

if and if .

In other words, represents the SIZE of (if you were to draw a number line and measure the distance from to ).

Now, since , that means that is something that has SIZE .

This means .

This is particularly useful if dealing with squares in inequalities.

Now back to the problem.

Let's find the value of for which there are not any solutions.

.

Now to undo this square, we need to take the square root:

But , so

.

So, this means that the "size" of has to be less than 1.

This means . You can verify this by drawing a number line (the distance of any of the points in this region from 0 is less than 1).

So for there to be no solutions, .

Finally, for 2 solutions:

.

So the size of has to be greater than 1.

This means or .

Once again, you can check this with a number line. The distance from any point or to is obviously going to be .

Therefore, for 2 solutions: or .

To summarise:

0 solutions: , which is the same as .

1 solution: , which is the same as or .

2 solutions: , which is the same as or .