How do you find the points of intersection of 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:

2x²=x²+1

This is a fairly easy equation to solve:

Lets make one side equal to zero:

-x² +1=0

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:

(x+1)(x-1)=0

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, 

y=2x²

y=2(1)² =2

When x= -1

y=2(-1)² = 2

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

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