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Graphical solutions to equations__

You already know how to solve quadratic
equations from the 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 x^{2} - 3x - 3 =
y is drawn, solving the equation x^{2} - 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 x^{2}
- 3x - 3 = 4, which is exactly the same as x^{2}
- 3x - 7 = 0 (taking 4 from both sides). So by
drawing the graph of y = x^{2} - 3x - 3 I can solve not only x^{2}
- 3x - 3 = 0 but also x^{2} - 3x - 7 = 0 etc.

Now suppose I want to solve x^{2}
- 5x - 8 = 0 using the graph of y = x^{2} - 3x - 3.

First I have to arrange the equation into
the form x^{2} - 3x - 3 = ???

x^{2} - 5x - 8 = 0

x^{2} - 5x - 3 = 5

*x*^{2} - 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 = x^{2} -
x - 4

Use the graph to solve the following
equations...

1) x^{2} - x - 4 = 0

2) x^{2} + 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) x^{2}
+ x - 7 = 0

x^{2} + x - 4 = 3

x^{2} - x - 4 = -2x + 3

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