1. Approximate Solution of Equations
Graphical Method
Example :
Solve the equation
x2 + 2x - 1 = 0

2. Method 1: Draw the graph y = x2 + 2x - 1

3. x = -2.4

4. Method 1: Draw the graph y = x2 + 2x - 1

5. x = 0.4

6. Method 2 : Use of the given graph y = x2
Rewrite the equation as x2 = -2x + 1
Add the line y = -2x + 1 on the same graph paper
Note the x-coordinates of the points of intersection

7. y = -2x+1
y = x2

8. Winplot Graphmatica http://math.exeter.edu/rparris
Drawing Software :
Winplot
Graphmatica

9. x2 – x – 1 = 0 x2 = x + 1 2x2 = 2x + 2 Answer to Worksheet Exercise 1:
Use the graph y = 2x2 to solve
x2 – x – 1 = 0
Rewrite the equation as
x2 = x + 1
2x2 = 2x + 2
Draw the line y = 2x + 2

10. -1 1 -2 2 3 4 5 6
y = 2x2 x y
y = 2x+2
(1.6, 5.3)
(-0.6, 0.7)
-0.6
1.6

11. Answer to Worksheet Exercise 1:
Use the graph y = 2x2 to solve
4x2 + x – 6 = 0
Rewrite the equation as
4x2 = -x + 6
2x2 = -x/2 + 3
Draw the line y = -x/2 + 3

12. -1 1 -2 2 3 4 5 6
y = 2x2 x y
y = -x/2+3
-1.4
1.1

13. Approximate Solution of Equations
Method of Bisection
Example :
Solve the equation
x3 - 3x2 + 5 = 0

14. The root lies between -2 and -1
Step 1 : Locate the root
y = x3 - 3x2 + 5
The root lies between -2 and -1

15. interval that contains the root
Step 2 : Find the mid-point of the
interval that contains the root -2 -1
Mid-point =
( ) = -1.5

16. Step 3 : Choose the half-interval
that contains the root
f(x) = x3 - 3x2 + 5 -2 -1
-1.5
-1.5 -1
-1.25

17. -1.1 Root of x3 - 3x2 + 5 = 0 is (1 d.p.) -1.25 -1 -1.125 -1.125 -1
Root of x3 - 3x = 0 is
-1.1
(1 d.p.)
-1.125

18. Choose the half-interval
Method of Bisection
Find the interval
that contains the root
Find the mid-point No
Precise
enough ?
Choose the half-interval
that contains the root
Answer
Yes

19. - Root of x3 - 3x2 + 5 = 0 is –1.1 (1d.p.) f(-2) = -15 f(-1) = 1 a b
-1.5
-5.13
-1.5 -1
-1.25
-1.64
-1.25 -1
-1.125
-0.22
-1.125 -1
0.42
-1.125
Root of x3 - 3x = 0 is –1.1 (1d.p.)

20. Use of Excel Spreadsheet
Method of Bisection
Use of Excel Spreadsheet

21. Bracketing interval : 1 < x0 < 2
Answer to Worksheet Exercise 3:
Find a root of 2x4 – 3x – 5 = 0
Bracketing interval : 1 < x0 < 2

22. The root is 1.5 (2 sig. fig.) a b 1 2 1.5 0.625 1.25 -3.87 1.375 -1.98
1.4375
-0.77
The root is (2 sig. fig.)

23. Answer to Worksheet Exercise 4:
Find a root of x3 – 7x +2 = 0

24. Useful Websites for
Method of Bisection
symbol/bisect.html