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hi bonus tutorial what I want to do is show you how we can establish that a root of an equation lies in a given interval but just as a reminder first of all what do we mean by roots of an equation well from a graphical point of view suppose weve got some curve y equals f of X these points where the curve crosses the x axis are called roots ok so these two points here then are essentially roots of this equation there are points where f of X equals 0 now curve can have two roots as you see in this example it can have just one root you can have many roots right ok might not even have any roots but what were going to be looking at is how do we know if a root lies in a given interval lets say weve got an interval here ok were at this point X is X 1 and at this point X is X 2 what what we do is we consider a change in sign by that I mean that if we were to put this value X 1 into the curve replace X with X 1 what we would be getting is this point down here which would be a negative val