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Graphical solutions to equations

You already know how to solve quadratic equations from the Algebra section. In this chapter we look at how to solve closely related quadratics without having to go through the method of factorising etc. each time.

When the graph of x2 - 3x - 3 = y is drawn, solving the equation x2 - 3x - 3 = 0 tells us what x is when y is zero (i.e. where it cuts the x axis). If you look at the expressions above, it is easy to see that we have simply replaced y with zero.

Now suppose that I want to know what x is when y = 4, i.e. where the curve crosses the line y = 4. I could simply draw the line y = 4 onto the graph above and read the answer off!

This is the solution to the equation x2 - 3x - 3 = 4, which is exactly the same as x2 - 3x - 7 = 0 (taking 4 from both sides). So by drawing the graph of y = x2 - 3x - 3 I can solve not only x2 - 3x - 3 = 0 but also x2 - 3x - 7 = 0 etc.

Now suppose I want to solve x2 - 5x - 8 = 0 using the graph of y = x2 - 3x - 3.

First I have to arrange the equation into the form x2 - 3x - 3 = ???

x2 - 5x - 8 = 0

x2 - 5x - 3 = 5

x2 - 3x - 3 = 2x + 5

Now I need to draw the line y = 2x + 5 on the graph just as I drew y = 4 and y = 0 (the x axis) before (I have also made the scale bigger so that the crossing points are on the graph).

The solutions to the equation are the x coordinates where the lines cross, which reads off to be about -1.2 and 6.3

 

Final example

Below is the graph of y = x2 - x - 4

Use the graph to solve the following equations...

1) x2 - x - 4 = 0

2) x2 + x - 7 = 0

 

Answer to 1)    'draw' the line y = 0 and read off. The solutions are about -1.6 and 2.6

Answer to 2)    x2 + x - 7 = 0

                         x2 + x - 4 = 3

                         x2 - x - 4 = -2x + 3

                         draw the line y = -2x + 3 and read off. The solutions are about -3.2 and 2.2

 

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