Quadratic inequalities ​

Notes ​

Define a parabola by

and let's take a look at a table of values:

What if we wanted to describe the set of all $x$ such that $y<0$? It might seem clear from the table, but what about $x=-1.5,x=\sqrt{2}$ and all the other non-integers? We can't check them all!

The easiest way to be sure is to sketch the parabola and look for when $y<0$ (that is, when the parabola lies below the $x$-axis)

Now it is clear that $y<0$ whenever $x>-3$ and $x<2.$

Example ​

Exercise ​