2. The general form of a quadaratic equations is ax²+bx+c=0,
where a,b and c are constant. The examples are:
a. 2x²-13x+15=0
b. k²-3k=0
c. 9m²-49=0
An equation should end with “=0”

3. A quadratic equation can be solved by 3
method:
a. Factorisation
b. Completing the square
c. Using a formula

4. a. Factorisation The curve shows
the way to expand two
brackets.
Example:
(3x+1)(x-1)=15 Equation should end with “=0”
3x²-3x+x-1=15
3x²-2x-1=15 This answer can get from
3x²-2x-1-15=0 calculator:
3x²-2x-16=0 Press “MODE” (3 times),
(3x-8)(x+2)=0 Press “1” for “EQN”,
x= 8 x= -2 Press the arrow button for
3 right (degree) and press “2”,
press 3 for a, -2 for b and -16
for c. You will get two values.

5. b. Completing the square
Example: When completing the square,
the equation should start
with “x²”
6x²-7x-3=0
6(x²-7x-3)=0
x²-7/6x-3/6=0
x²-7/6x-1/2=0 x-7/12= -11/12
x²-7/6x=1/2 x= -1/3
x²-7/6x+49/144=1/2+49/144
(x-7/12)²=(72+49)/144 x-7/12=11/12
(x-7/12)²=121/144 x=3/2
x-7/12=±√121/144
x-7/12=±11/12

6. c.Using a formula Formula: x= -b±√b²-4ac
2a
Example:
18x²+27x-35=0
So, x= -27±√27²-4(18)(-35)
2(18)
x= -27±√3249
a is 18 36
b is 27 x= -27±57
c is -35 36
x= -27 + 57
36
x= -27 – 57
36
x= 5
6
x= -7
3

7. -b
When forming a quadratic equation from a
roots:
C
a
1. Calculating the “sum of roots”
2. Calculating the “product of roots”
3. Forming a quadratic equation by:
x²- (sum of roots)x + (product of
roots)=0

8. The sum of roots and product of roots for the quadratic equations:
Example:
Sum of roots: -b
a
3x²+5x-9=0
-5
3
So a is 3, b is 5 and c is -9
Product of roots: c
a
-9
3
-3

9. TYPES OF ROOTS BASED ON THE CONDITIONS:
b²-4ac>0 b²-4ac=0 b²-4ac<0 b²-4ac ≥0
Two real and Two real and
No real roots Two real roots
distinct roots equal roots

10. Example :
Find the range of values of p if the quadratic equation 2x²-4x=2x-3+px² has two real and
distinct roots.
2x²-4x=2x-3+px²
2x²-px²-6x+3=0
Change into general form
(2-p)x²-6x+3=0
ax²+bx+c
a=2-p
b= -6
c= 3
Apply the formula. Since it is
(-6)²-4(2-p)(3)>0 two real and distinct roots,
36-12(2-p)(3)>0 apply b²-4ac>0
36-24+12p>0
12+12p>0
12p>-12
p> -1