How do I find the points of intersection between two curves?

This problem is a graphical representation of finding the solutions to a pair of simultaneous equations.

In this example we will use the curves y=2x2 , and y=x2+1. This is a very straightforward example, but demonstrates the method of finding the intersection of two curves well.

Step 1  - since the LHS of both these equations is the same (y=...) we can equate the two equations:


This is a fairly easy equation to solve:

Lets make one side equal to zero:


Lets move everything across to the other side to get rid of the minus signs.

x2 - 1 =0

This is the difference of two squares, so can be factorised:


So the x-coordinates of the intersection points are  +1 and -1.

Step 2 - Now we need to find the y-coordinates. We do this by plugging the x-values into the original equations. We can use either one, because the lines intersect (so they should give us the same result!)

When x= +1, 



When x= -1

y=2(-1)= 2

So the points of intersection have coordinates (-1,2) and (1,2)

We can see this graphically: (see how easy this example was!)

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