2.2 Solution of an Equation by Graphical Method


2.2 Solving equations graphically

The solution of the equation f (x) = g (x) can be solved by graphical method.
Step 1: Draw the graphs of y = f (x) and y = g (x) on the same axes.
Step 2: The points of intersection of the graphs are the solutions of the equation
f (x) = g (x). Read the values of x from the graph. 



Solution of an Equation by Graphical Method

Example 1:
(a) The following table shows the corresponding values of x and y for the equation y = 2x2 x – 3.
 
x
–2
–1
–0.5
1
2
3
4
4.5
5
y
7
m
– 2
–2
3
12
n
33
42

Calculate the value of m and n.

(b) For this part of the question, use graph paper. You may use a flexible curve rule.
By using a scale of 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis, draw the graph of  y = 2x2 x – 3 for –2 ≤ x ≤ 5.

(c) From your graph, find
(iThe value of y when x = 3.9,
(iiThe value of x when y = 31.

(d) Draw a suitable straight line on your graph to find the values of x which satisfy the equation 2x2 3x = 10 for –2 ≤ x ≤ 5.

Solution:
(a)
y = 2x2 x – 3
when x = –1,
m = 2 (–1)2 – (–1) – 3 = 0
when x = 4,
n = 2 (4)2 – (4) – 3 = 25

(b)

(c)
(i) From the graph, when x = 3.9, y = 23.5
(iiFrom the graph, when y = 31, x = 4.4

(d)
y = 2x2 x – 3 ----- (1)
2x2 3x = 10 ----- (2)

y = 2x2 x – 3 ----- (1)
0 = 2x2 3x – 10 ------ (2) ← (Rearrange (2))
(1)  – (2) : y = 2x + 7
The suitable straight line is y = 2x + 7.

Determine the x-coordinates of the two points of intersection of the curve 
y = 2x2 x – 3 and the straight line y = 2x + 7.

x
0
4
y = 2x + 7
7
15

From the graph, x = –1.6, 3.1