Basic Engineering Mathematics, Fifth Edition

Graphical solution of equations 161

 4

 21  1 0

B

A

y 2  4 x

y 2 x^2  3 x 4

23 x

2

4

6

8

10

y

 2

Figure 19.13

Now try the following Practice Exercise

PracticeExercise 74 Solving linear and

quadratic equations simultaneously

(answers on page 348)

1. Determine graphically the values ofx and
   ywhich simultaneously satisfy the equations
   y= 2 (x^2 − 2 x− 4 )andy+ 4 = 3 x.


2. Plot the graph ofy= 4 x^2 − 8 x−21 for values
   ofxfrom−2to+4. Use the graph to find the
   roots of the following equations.
   (a) 4x^2 − 8 x− 21 = 0
   (b) 4x^2 − 8 x− 16 = 0
   (c) 4x^2 − 6 x− 18 = 0

19.4 Graphical solution of cubic equations

Acubic equationof the formax^3 +bx^2 +cx+d= 0
may be solved graphically by

(a) plotting the graphy=ax^3 +bx^2 +cx+d,and

(b) noting the points of intersection on thex-axis (i.e.
wherey=0).

The x-values of the points of intersection give the

required solution since at these points bothy=0and

ax^3 +bx^2 +cx+d=0.

The number of solutions, or roots, of a cubic equation

depends on how many times the curve cuts thex-axis

and there can be one, two or three possible roots, as

shown in Figure 19.14.

(a)

y

x

(b)

y

x

(c)

y

x

Figure 19.14

Here are some worked problems to demonstrate the

graphical solution of cubic equations.

Problem 8. Solve graphically the cubic equation

4 x^3 − 8 x^2 − 15 x+ 9 =0, given that the roots lie

betweenx=−2andx=3. Determine also the

co-ordinates of the turning points and distinguish

between them

Lety= 4 x^3 − 8 x^2 − 15 x+9.Atableofvaluesisdrawn

up as shown below.

x − 2 − 1 0 1 2 3

y − 25 12 9 − 10 − 21 0

A graph of y= 4 x^3 − 8 x^2 − 15 x+9isshownin

Figure 19.15.

Thegraphcrossesthex-axis(wherey=0)atx=− 1. 5 ,

x= 0. 5 andx= 3 and these are the solutions to the

cubic equation 4x^3 − 8 x^2 − 15 x+ 9 =0.

The turning points occur at(− 0. 6 , 14. 2 ),whichisa

maximum,and( 2 ,− 21 ),whichisaminimum.

Problem 9. Plot the graph of

y= 2 x^3 − 7 x^2 + 4 x+4 for values ofxbetween

x=−1andx=3. Hence, determine the roots of

the equation 2x^3 − 7 x^2 + 4 x+ 4 = 0

A table of values is drawn up as shown below.

x − 1 0 1 2 3

y − 9 4 3 0 7