1. Solving Quadratic Equations by the Graphical Method

2. Miss Chan, how to plot a quadratic graph like y = 2x2 + 5x  7?
You can choose a range of values of x and find the corresponding value of y.

3. Consider the graph of y = 2x2 + 5x  7.
Step 1
Find the value of y corresponding to each integral value of x from x = 4 to x = 1. x 4 3 2 1 1 y 5 4 9
10 7
Step 2
Plot the point with coordinates (x, y) for each pair of x and y in the table.

4. Join these points with a smooth curve.
Step 3 y
x-intercept 4
x-intercept
Join these points with a smooth curve. 2
(–3.5, 0.0)
(1.0, 0.0) x 3 2 1
The corresponding value of y is 0.0 when x equals –3.5 and 1.0. 4 1 2 4
y = 2x2 + 5x  7 6 8
parabolic shape
10

5. x-intercepts of y = 2x2 + 5x  7 are 3.5 and 1.0
2x2 + 5x  7 = 0 when x = 3.5 or 1.0
3.5 and 1.0 are the roots of 2x2 + 5x  7 = 0
The x-intercepts of the graph of y = ax2 + bx + c (where a ≠ 0) are the roots of ax2 + bx + c = 0.

6. When finding the roots of ax2 + bx + c = 0 by the graphical method, there are three possible cases:
No. of
x-intercepts of the graph of
y = ax2 + bx + c
The graph has two x-intercepts.
e.g.
The graph has only one x-intercept.
e.g.
The graph has no
x-intercepts.
e.g. y x y x y x
Nature of roots of
ax2 + bx + c = 0
2 unequal real roots / 2 distinct real roots
It refers to whether the roots are
1. real or not real,
2. equal or unequal.
2 equal real roots / 1 double real root
no real roots

7. Follow-up question Find the roots of the quadratic equation
x2  2x + 3 = 0.
The x-intercepts are 3 and 1. y 4 2 2 4
Therefore, the roots of x2  2x + 3 = 0 are 3 and 1.
y = x2  2x + 3 x
4 3 2 1 1 2
Note: We can only find the approximate values of the roots by the graphical method.