162 Basic Engineering Mathematics
22
21 20.6
24
4
8
12
16
y
14.2
28
212
216
220
221
224
0
1
2
3
x
y 5 4x 3 2 8x 2 2 15x 1 9
Figure 19.15
A graph of y = 2x 3 − 7x 2 + 4x + 4 is shown in
Figure 19.16. The graph crosses the x-axis at x = −0.5
and touches the x-axis at x = 2.
28
26
24
22
21
1
2
3
0
2
4
6
8
y
x
y 5 2x 3 2 7x 2 1 4x 1 4
Figure 19.16
Hence the solutions of the equation
2x 3 − 7x 2 + 4x + 4 = 0 are x = −0.5 and x = 2.
Now try the following Practice Exercise
Practice Exercise 75 Solving cubic
equations (answers on page 348)
1. Plot the graph y = 4x 3 + 4x 2 − 11x − 6
between x = −3 and x = 2 and use
the graph to solve the cubic equation
4x 3 + 4x 2 − 11x − 6 = 0.
2. By plotting a graph of y = x 3 − 2x 2 − 5x + 6
between x = −3 and x = 4, solve the equation x 3 − 2x 2 − 5x + 6 = 0. Determine also
the co-ordinates of the turning points and
distinguish between them.
In problems 3 to 6, solve graphically the cubic
equations given, each correct to 2 significant
figures.
3. x 3 − 1 = 0
4. x 3 − x 2 − 5x + 2 = 0
5. x 3 − 2x 2 = 2x − 2
6. 2x 3 − x 2 − 9.08x + 8.28 = 0
7. Show that the cubic equation
8x 3 + 36x 2 + 54x + 27 = 0 has only one real
root and determine its value.