I don't understand how to solve quadratic inequalities?
Let us look at an example so we can see what is going on as we go through the method. Be cautious of just memorising the method; understanding it will make your life far easier.
Q: x2 + 2x - 8 < 0
Intuition: When we are looking at quadratic equations, it is easy to get confused. When you usually solve them, you find two roots because the graph is parabolic in shape instead of a straight line. Therefore, when we try and solve an inequality, the region we want to find is not immediately obvious. The first step is to visualise what is going on. If we draw x2 + 2x - 8 = y (which we can do by substituing a few values in for x and plotting it), we get the graph we are looking at. Now the question is asking us for the region where the y is below zero. We can see that that is between the two points where y intercepts the x-axis (or the y = 0 line). Therefore, all we need to do is find the intersection and our answer is the region between those two points!
Method:
1) Find the points of intersection with the line where the equality holds. In this case, it is the y = 0 line (the x-axis) so we are looking to solve x2 + 2x - 8 = 0. We do this by factorisation (find two numbers that mulltiply to make -8 and add to make 2). The answer is (x+4)(x-2) = 0 and therefore, x = -4 and x=2
2) Find what region you want using the graph. In this case, we want y to be smaller than 0, and from the graph we can see that this happens between the two values. Therefore, our solution is -4 < x < 2.
