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 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
Below is the graph of y = x2 -
x - 4
Use the graph to solve the following
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