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ADDITIONAL MATHEMATICS
FORM 4
QUADRATIC EQUATIONS<= /div>
FORM 4
QUADRATIC EQUATIONS<= /div>
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Exercise
Note<=
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Example
Answ=
er<=
b>
2 :
QUADRATIC EQUATIONS
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2.1 Quadratic Equation & Its Root=
s
•The genaral form of Quadratic Equation
•ax2 + bx + c =3D 0
•a ≠ 0
<=
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<=
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lang=3DMS style=3D'mso-special-format:lastCR;display:none'>
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2.2
Solving a Quadratic Equation
•<=
b>1. A
quadratic equation can be solved by
•<=
span
style=3D'mso-spacerun:yes'> =
;
(a) Factorisation
•<=
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style=3D'mso-spacerun:yes'> =
;
(b) Completing the square
<=
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class=3DHBB style=3D'position:absolute;left:-4.85%'>• =
;
(c) Using a formula
<=
span
style=3D'visibility:hidden'>•
<=
span
class=3DHBB style=3D'position:absolute;left:-4.85%'>•2. The formula :
<=
span
class=3DHBB style=3D'position:absolute;left:-4.85%'>• =
;
x =3D
<=
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lang=3DMS style=3D'mso-special-format:lastCR;display:none'>
<=
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lang=3DMS style=3D'mso-special-format:lastCR;display:none'>
3. Using S.O.R and P.O.=
R
x2 - (S.O.R)x + (P.O.R) =3D 0
S.O.= R =3D sum of roots P.O.R =3D produ= ct of roots
x2 - (S.O.R)x + (P.O.R) =3D 0
S.O.= R =3D sum of roots P.O.R =3D produ= ct of roots
•
•
•
<=
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lang=3DMS style=3D'mso-special-format:lastCR;display:none'>
2.3 Types of the roots of the Quad=
ratic Equation
•(=
a) Two
real and distinct roots :b2-4ac >0
•(b) Two real and eq=
ual
roots : b2 – 4ac =3D=
0
•(=
c) No
real roots : b2 – 4ac <=
; 0
•(=
d) Two
real roots : b2 – 4ac ≥ 0
<=
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lang=3DMS style=3D'mso-special-format:lastCR;display:none'>
<=
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lang=3DMS style=3D'mso-special-format:lastCR;display:none'>
=
Example
1 : Express each of =
the
following
= ; in the general form
= ; &n= bsp; (x + 1 )2&nbs=
p;=3D
16
= ; in the general form
= ; &n= bsp; (x + 1 )2
• =
;
(x + 1 )2 =3D 16
• =
;
x2 + 2x + 1 =3D16
• =
;
x2 + 2x + 1 – 16 =3D 0
• =
;
x2 + 2x – 15 =3D 0
![](3D"slide0004_image019.gif")
![](3D"slide0004_image021.gif")
=
Example 2 : Solve the following quadratic
= ; equation by factorisation.
= ; x2 + 7x= + 10 =3D 0
= ; equation by factorisation.
= ; x2 + 7x= + 10 =3D 0
<=
span
style=3D'visibility:hidden'>•
• =
;
x2 + 7x +1=
0 =3D 0
• =
;
(x + 2) (x + 5) =3D 0
• =
;
x =3D -2 or x =3D -5
![](3D"slide0020_image023.gif")
=
=
= Example 3 : Solve the following quadratic
= ; equation by quadratic formula.
= ; 3x2&nbs=
p;+
6x + 2 =3D 0
=
= Example 3 : Solve the following quadratic
= ; equation by quadratic formula.
= ; 3x2
• =
;
a =3D 3 , b =3D 6 ,
c =3D 2
<=
span
style=3D'visibility:hidden'>•
<=
span
style=3D'visibility:hidden'>•
<=
span
style=3D'visibility:hidden'>•
<=
span
style=3D'visibility:hidden'>•
• =
;
x =3D -0.4226 x =
=3D
-1.577
<=
span
style=3D'visibility:hidden'>•
![](3D"slide0018_image027.gif")
=
Example 4 : Solve the following quadratic
= ; equation by completing the
= ; square.
= ; x2 + 8x= - 6 =3D 0
= ; equation by completing the
= ; square.
= ; x2 + 8x= - 6 =3D 0
=
;
x2 + 8x - =
6 =3D 0
=
;
x2 + 8x =
;
=3D 6
• =
;
(x + 4)2 =
;
=3D=
&nbs=
p;
6 +=
42
=
; &n=
bsp; =
;
=3D 22
• =
;
x + 4 =
;
=3D +
• =
; &n=
bsp;
x =3D - 4 +
=
=
; &n=
bsp; =
;
=3D 0.6904 or
-8.6904
•
![](3D"slide0021_image030.gif")
=
Example 5 : Form a quadratic equation with
= ; the roots -2 and 4
= ; the roots -2 and 4
• =
;
S.O.R =3D -2 + 4 =3D 2
• =
;
P.O.R =3D -2 x 4 =3D =
-8
• =
;
The quadratic equation is
• =
;
x2 – (2)x + (-8) =3D 0
=
;
x2 – 2x – 8 =3D 0
![](3D"slide0022_image034.jpg")
=
Example 6 : If m and n are the roots of the
= ; equation 2x2&nbs=
p;+
8x – 3 =3D 0 ,
=
;
form the equation whose roots
= ; are 2m and 2n .
= ; equation 2x2
= ; are 2m and 2n .
=
;
2x2&nbs=
p;+
8x – 3 =3D 0
m + n =3D -8/2 =
=3D
-4
mn =3D -3/2 =
3;
=
S.O.R =3D 2m + =
2n
P.O.R =3D (2m)(2n)
=
;
=3D 2 (m+n) =
;
=3D 4mn
=
;
=3D 2(-4) =
; &n=
bsp;
=3D 4(-3/2)
=
=
;
=3D -8 =
; &n=
bsp;
=3D -6
=
=
; &n=
bsp;
x2-(-8x) + (-6) =3D 0
=
; &n=
bsp;
x2 + 8x – 6 =3D 0 =
;
![](3D"slide0023_image035.jpg")
=
Example 7 : Determine the types of roots
= ; for each of the following
= ; quadratic equation
=
;
x2 - 12=
x + 27 =3D
0
= ; for each of the following
= ; quadratic equation
=
;
x2 - 12x + 27 =3D 0
=
;
a =3D 1 , b =3D -12 , c =3D 27
=
;
b2 R=
11; 4ac
=3D (-12)2 R=
11;
4(1)(27)
Thus, x2 - 12x + 2 =3D 0 has two distinct roots.
=
=
;
![](3D"slide0024_image037.gif")
=
Example 8 : The quadratic equation =
;
= ; x2 R= 11; 2px + 25 =3D 0 has two
= ; equal roots.
= ; x2 R= 11; 2px + 25 =3D 0 has two
= ; equal roots.
• =
;
x2 -2px + 25 =3D 0=
• =
;
a =3D 1 , b =3D -2p , c =3D 25
• =
;
b2- 4ac =3D 0
<=
/span>
• =
;
(-2p)2 – 4(1)(25=
) =3D
0
• =
;
4p2 – 100 =
;
=3D 0
• =
;
p2 =3D 25
• =
;
p
=3D +
=
;
• =
;
p
=3D +<=
/span> 5
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