Basic Engineering Mathematics, Fifth Edition

162 Basic Engineering Mathematics

22 212 0.6

24

4

8

12

16

y

14.2

28

212

216

220

221

224

0123 x

y 54 x^328 x^2215 x 19

Figure 19.15

A graph of y= 2 x^3 − 7 x^2 + 4 x+4isshownin

Figure 19.16. The graph crosses thex-axis atx=− 0. 5

and touches thex-axis atx=2.

28

26

24

22

210 123

2

4

6

8

y

x

y 52 x^327 x^214 x 14

Figure 19.16

Hence the solutions of the equation

2 x^3 − 7 x^2 + 4 x+ 4 =0arex=− 0. 5 andx= 2.

Now try the following Practice Exercise

PracticeExercise 75 Solving cubic

equations (answers on page 348)

1. Plot the graph y= 4 x^3 + 4 x^2 − 11 x− 6
   between x=−3andx=2 and use
   the graph to solve the cubic equation
   4 x^3 + 4 x^2 − 11 x− 6 =0.


2. By plotting a graph ofy=x^3 − 2 x^2 − 5 x+ 6
   betweenx=−3andx=4, solve the equa-
   tionx^3 − 2 x^2 − 5 x+ 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 − 5 x+ 2 = 0


5. x^3 − 2 x^2 = 2 x− 2


6. 2x^3 −x^2 − 9. 08 x+ 8. 28 = 0


7. Show that the cubic equation
   8 x^3 + 36 x^2 + 54 x+ 27 =0 has only one real
   root and determine its value.