1. Ronald Hui Tak Sun Secondary School
Mathematics
Ronald Hui
Tak Sun Secondary School

3. 2016 HKDSE
Ronald HUI

4. 2017 HKDSE
Ronald HUI

6. Follow-up question Factorize x2 + 2x – 15. Solution
List all the possible pairs of factors of –15.
Check each pair of factors by the cross-method as follows:

7. Quadratic Equations in One Unknown

8. highest degree of the unknown is 2
What is quadratic equation in one unknown?
x 2  5x + 6 = 0, 2
8x 2 + 2x  3 = 0, 2
highest degree of the unknown is 2
one unknown x
4y 2 = y + 1, 2
5y 2 = 20 2
one unknown y
These equations are all quadratic equations in one unknown.

9. a, b and c are real numbers
General form of quadratic equations in one unknown
ax2 + bx + c = 0
a  0
a, b and c are real numbers
If a = 0, the equation becomes a linear equation bx + c = 0.
Rearranging terms
For example:
(i) 2x2  5x + 6 = 0 2
 5
+ 6
(ii) 2x  3 + x2 = 0
We usually keep a positive.
x2 + 2x  3 = 0 1
+ 2
 3 a b c a b c

10. In fact, all quadratic equations can be written in general form.
Rewrite the following quadratic equations in general form.
By transposing terms,
2x2 + 5x  3 = 0
2x2 + 5x = 3
By expanding the equation,
x2 + x + 3 = 0
x(x + 1) + 3 = 0
By expanding and transposing terms, x2  4x + 3 = 0
(x  2)2 = 1

11. What is a root of a quadratic equation ax2 + bx + c = 0?
Roots of a quadratic equation
What is a root of a quadratic equation ax2 + bx + c = 0?
A root of an equation is a value of x that satisfies the equation.
For example, 3 is a root of the equation x2  9 = 0.
 32 – 9 = 0

12. Is 1 a root of the equation x 2  5x  6 = 0?
Follow-up question
Is 1 a root of the equation x 2  5x  6 = 0?
Substitute x = 1 into the equation.
L.H.S.
= (1)2  5(1)  6
= 0
= R.H.S.
 1 is a root of the equation x 2  5x  6 = 0.

13. Solving Quadratic Equations by Factor Method

14. For any expression in the form (px  r)(qx  s),
For any two real numbers a and b, if
(px  r)(qx  s) = 0
if ab = 0
a = 0 or
b = 0
px  r = 0 or
qx  s = 0

15. How to solve a quadratic equation using the factor method?
ax2 + bx + c = 0
(px  r)(qx  s) = 0
factorize ax2 + bx + c
px  r = 0 or qx  s = 0
x = or x = r p s q
The roots of ax2 + bx + c = 0 are and . r p s q

16. x2 + x  12 = 0 using the factor method?
Can you solve
x2 + x  12 = 0 using the factor method?
x2 + x  12 = 0
Factorize
x2 + x  12 first.
(x  3)(x + 4) = 0 x –3 +4
–3x
+4x = +x
x  3 = 0
or x + 4 = 0
x = 3
or x = 4
 The roots of x2 + x  12 = 0 are 3 and 4.

17.  The roots of 2x2  x  6 = 0 are and 2.
Follow-up question
Solve 2x2  x  6 = 0 using the factor method.
2x2  x  6 = 0
Factorize 2x2 – x  6.
(2x + 3)(x  2) = 0
2x + 3 = 0
or x  2 = 0 3
x =  2
or x = 2
 The roots of 2x2  x  6 = 0 are and 2. 3 2