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y dditionalMathematicso uleForm 4(Version 2010)Topic 2:
Quadratic
Equationsby
NgKL(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH)
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2.1 QUADRATIC EQUATION AND ITS ROOTSPERSAMAAN KUADRATIK DAN PUNCANYA
Quadratic Equations in General For !Bentuk Am Persamaan Kuadratik)
1. The general form of a quadratic equation is given by 02 =++ cbxax where a, b, and c
are constants, 0a andxan unknown.Bentuk a bagi persaaan kuadratik ada!a" 02 =++ cbxax di ana a, b, dan c sebagai pea!ar, 0a dan xsebagai anu.
Examles of quadratic equations in general formConto"#conto" persaaan kuadratik da!a bentuk a ia!a"!
$x%& '%x * + k%& %k * + -%& % * +
2. "bserve the examles, the highest degree #ower$ of the unknown of the quadratic
euations is 2.Per"atikan conto" di atas, kuasa tertinggi anu persaaan kuadratik ia!a" 2.
%. The roots of a quadratic equation are values of the unknown that satisfy the equation.
Punca persaaan kuadratik ada!a" ni!ai bagi anu ang euaskan persaaan itu.
&. ' quadratic equation can only has the highest of two roots.(ersamaan kuadratik memunyai selebih)lebihnya dua unca saha*a.
+. The rimitive method to determine the roots of a quadratic equation is by substitution or
trial and error methodCara ang pa!ing priiti/ untuk enentukan punca#punca suatu persaaan kuadratik ia!a" dengan kaeda" penggantian
dan peerinuan (kaeda" cuba#cuba).
E"ercise 2.1
1. rite each of the following quadratic equation in general form.0u!iskan setiap persaaan kuadratik berikut da!a bentuk a.
#a$ +$2# =+xx #b$ $+2#%$ xxxx =
#c$ 1%$%#2 2 =+x
#d$ -2
2 =x
x
2. rite whether the value given in each of the following quadratic equations is the root of
the quadratic equation.0entukan saa ada ni!ai ang diberikan ia!a" punca bagi persaaan kuadratik berikut
#a$ &0&+2 ==+ xxx
#b$%
102/% 2 ==++ xxx
#c$+2-1/+ 2 == xxx #d$
-11$/-# == xxx
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x
x
%x
)-x
2x
%
)-
)1 )%x
x
2x
)-x
+x
2
2x
)%
+
)1+ )x
2.2 SO#UTION o$ QUADRATIC EQUATIONS(PENYELESAIAN PERSAMAAN KUADRATIK)
1. To solve a quadratic equation means to find the roots of the quadratic equation.(Mene!esaikan suatu persaaan kuadratik bererti encari punca#punca bagi persaaan kuadratik itu).2. enerally, there are threes methods to determine the roots of a quadratic equation
02 =++ cbxax Secara ana terdapat tiga cara da!a enentukan punca suatu persaaan kuadratik 02 =++ cbxax #a$ 3actorisation, (Pe/aktoran)
#b$ 4omleting the square, (Penepurnaan 1uasa Dua)
#c$ 5uadratic 3ormula. (2uus kuadratik)
!A% Solution &' Factorisation (Pene!esaian se"ara Pem#akt$ran).
1. To determine the roots of a quadratic equation 02 =++ cbxax , factor comletely the
exression cbxax ++2
to the form (x p)(nx 3) 4it" , n, p and 3 are constants.(5ntuk enentukan punca persaaan kuadratik berbentuk 0
2 =++ cbxax , /aktorkan se!engkapna ungkapankuadratik cbxax ++
2 kepada bentuk $$## 3nxpx ++ dengan , n, p dan 3 sebagai pea!ar).
E"a(le 1.
6olve each of the following quadratics equations.
(Se!esaikan setiap persaaan kuadratik ang berikut)
#a$ 1%2 = xx#b$ 12$%2$#1# =+ xx
6olution! (Pene!esaian)
#a$ 1%2 = xx
0$-$#%#01% 2
=+=
xxxx
Therefore, (Maka), 0-0% ==+ xorx %=x -=x
6olution! (Pene!esaian)
#b$ 12$%2$#1# =+ xx
0$+2$#%#01+212%2
2
2
=+==
xxxxxx
Therefore, (Maka), +)%xorx =+= 0%
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%=x 2
+=x
6olve each of the following quadratic equation by factori7ation.(Se!esaikan setiap persaaan kuadratik berikut dengan enggunakan keada" pe/aktoran).
#a$ 0-2 =xx #b$ 0&01%2 =++ xx
#c$ 01+2 =++ xx #d$ 0&/2 2 =+ xx
#e$ 0&/1+ 2 =+ xx #f$ 0%10 2 =+ xx
#g$ 0%221
2
= xx #h$ 0/&1- 2
=+ xx
E"ercise 2.2)
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#i$ 0%2 2 = xx #*$ 022110 2 =+ xx
!*% Solution &' Co(letin+ t,e Square -et,od!en'elesaian secara en'e(urnaan /uasa Dua%
1. 4omleting the square method is most suitable to be used if factori7ation method cannot
be emloyed or when the values of a, b andcare large.
#1aeda" aat sesuai digunakan 6ika keada" pe/aktoran tidak dapat di/aktorkan atau ni!ai a, b dan c da!a persaaankuadratik agak besar).
2. #a$ To do a comleting the square to the exression of bx%ax + , the term2
2
ab is
added u to the exression ax% bx.
89ngkaan bxax +2 boleh di*adikan kuasa dua semurna dengan menambahkan sebutan2
2
a
b
:.
22
2
22
2
22
22
22
+=+
++=
+=+
a
b
a
b
xbxax
a
b
a
bx
a
bx
xa
bxbxax
#b$ 6imilarly, to do a comleting the square to the exression ax% bx c , the term2
2
a
bis added u to the exression ax% bx c
8 ;eadaan yang sama, ungkaan cbxax ++2 boleh di*adikan kuasa dua semurna dengan menambahkan
sebuatan
2
2
a
b :
22
22
2
22
22
22
+
+=
+
++=
++=++
a
b
a
c
a
bx
a
c
a
b
a
bx
a
bx
a
cx
a
bxcbxax
E"a(le 2)
Exress the following quadratic exression in the form of comleting the square.(5ngkapkan ungkapan kuadratik berikut da!a bentuk kuasa dua sepurna).
#a$ xx &2 + #b$ xx %2 2
6olution! #(enyelesaian$!
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&$2#$2#$2#&
2
&
2
&&&$#
2
222
22
22
+=++=
++=+
x
xx
xxxxa
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