quadratic eq n 12
TRANSCRIPT
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y
dditionalMathematics
o ulesForm 4
(Version 2012)
Topic 2:
Quadratic
Equations byNgKL
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IMPORTANT NOTES
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x
x
+3x
-6x
2 x
+3
-6
-18 -3x
3. 2.1 QUADRATIC EQUATION AND ITS ROOTS ( PERSAMAAN KUADRATIK DAN PUNCANYA)
1. Write each of the following quadratic equation in general form . !"u#is$%n seti%p pers%&%%n $u%dr%ti$ 'eri$ut d%#%& 'entu$ %&(.
(a) 5)2( =+ x x (b) )52(3)4( x x x x =
(c) 13)3(2 2 =+ x (d) 62
2 = x x
1. Write whether the value given in each of the following quadratic equations is the root of the quadratic equation. ("entu$%n s%&% %d% ni#%i )%ng di'eri$%n i%#%* punc% '%gi pers%&%%n $u%dr%ti$ 'eri$ut(.
(a) 4!452 ==+ x x x(b)
31
!2"3 2 ==++ x x x
(c) 52
61"5 2
== x x x (d) 61
1)"6( == x x x
2.2 SOLUTION of QUADRATIC EQUATIONS !PEN E ES N PE/S M N 0 D/ " 0(
#o solve a quadratic equation $eans to find the roots of the quadratic equation. Men)e#es%i$%n su%tu pers%&%%n $u%dr%ti$ 'ererti &enc%ri punc%-punc% '%gi pers%&%%n $u%dr%ti$ itu.
1. %enerally& there are threes $ethods to deter$ine the roots of a quadratic equation !2 =++ c'x%x Sec%r% %&n)% terd%p%t tig% c%r% d%#%& &enentu$%n punc% su%tu pers%&%%n $u%dr%ti$ %x2 + 'x + c 4.5
(a) 'actorisation& !Pe& %$tor%n((b) o$ leting the square& !Pen)e&purn%%n 0u%s% Du%((c) *uadratic 'or$ula. !/u&us $u%dr%ti$(
(A) Sol t!on "# $a%tor!&at!on (Penyelesaian secara Pemfaktoran)
E'am le +olve each of the quadratics equation& 183x2 x = . Se#es%i$%n seti%p pers%&%%n $u%dr%ti$ )%ng 'eri$ut .
x2 7 3x 18 x2 3x 7 18 4
!x + 3(!x 7 6( 4
#herefore& !M%$%(, x + 3 4 or x 7 6 4
x 3, x 6
E'er%!&e 2.1.1
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4.
1. +olve each of the following quadratic equation by factori,ation. Se#es%i$%n seti%p pers%&%%n $u%dr%ti$ 'eri$ut deng%n &enggun%$%n $e%d%* pe& %$tor%(.
(a) !62 = x x (c) 9x2 + x 7 : 4
(b) !15-2 =++ x x (d) !4"2 2 =+ x x
(*) Sol t!on "# Com let!ng t+e S, are Met+o- (Penyelesaian secara Penyem!"rnaan K"asa D"a)
E'am le
+olve the following quadratic equation by co$ leting the square. (+elesai an ersa$aan uadrati beri ut secara enye$ urnaan uasa dua).
E'er%!&e 2.1.2
(b) 2 x2 + 3x :
2:
x23
x =+2
x2 +23
x +2
+=
:
32
2
:
3
2
+:3
x 2 +2
:3
16
;32 +
16
:1
16 :1
x =
+
43
1641
43 = x
1641
43 += x
-5!-.!= x < or
1641
43 = x
391.2 x = <
(a) x2 7 3x 7 9 4 x2- 3x 9
x2 3x +2
+=
2
39
2
2
3
2
23
x 9+2
23
:
;24 +
:
2;
:
2; x =
23
4
2/23 = x
:2;
23
x +=
1/3.4= x < or
4
2/
2
3 = x
1/3.1= x <
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5.
+olve the following quadratic equation by co$ leting the square $ethod.!Se#es%i$%n pers%&%%n $u%dr%ti$ 'eri$ut deng%n $%ed%* pen)e&purn%%n $u%s% du%(.
(a) !462 =++ x x (b) !31!2 = x x
(c) x x 453 2 = (d) )1(32 2 += x x
(e) !142 2 =+ x x (f) x3 x3: (2 x! 2 2 =++
E'er%!&e 2.1.
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6.
(C) Sol t!on "# Q a-rat!% $orm la (Pen#ele&a!an &e%ara r m & / a-rat!/)
#he quadratic for$ula is obtained by co$ leting the square $ethod as shown below. ! /u&us $u%dr%ti$ dipero#e*i deng%n $%ed%* pen)e&purn%%n $u%s% du% pers%&%%n $u%dr%ti$ seperti ditun=u$$%n di '%>%*(.
*uadratic for$ula%
%c'' x
242 =
can be use to solve any quadratic equationeven though the equation can be solve byeither factorisation or co$ leting the square$ethods.
(0u$us uadrati %
%c'' x
2
42
= boleh diguna an
untu $enyelesai an sebarang ersa$aan uadrati tan a$engira sa$a ada ersa$aan itu boleh diselesai an dengan$enggunaan aedah e$fa toran dan enye$ urnaanuasa dua atau tida ).
+olve each of the following quadratic equations by using the quadratic for$ula!Se#es%i$%n seti%p pers%&%%n $u%dr%ti$ 'eri$ut deng%n &enggun%$%n $%ed%* ru&u(
(a) 452 2 =+ x x (b) 411 x:2 x3 =++
%2%c:''
x
%2%c:'
%2'
x
%:
%c:'
%c
%:
'%2
' x
%2'
%c
%2'
x%'
x
%c
x%'
x
4%c
x%'
x
2
2
2
2
2
22
222
2
2
=
=+
=
=
+
+=
++
=+
=++!2 =++ c'x%x E'am le
+olve the quadratic equation 4 2 - 1 ! using
quadratic for$ula.(+elesai an ersa$aan uadrati !1-4 2 =+ x x secararu$us uadrati ).
+olution7 (8enyelesaian).
!1-4 2 =+ x x1&-&4 === c'%
9sing quadratic for$ula&%
%c'' x
242 =
8:88
816 6:8
(:! 2
(1 (! :! : (8! (8! x
2
=
=
=
#herefore&-
/2-.6--
4-- == x
866 .1 x8;28.1:
8;28.6 8
x
==
+=
or
134.!
-!"2.1
-/2-.6-
=
=
=
x
x
E'er%!&e 2.1.0
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3.
".
-.
(c) 3-2 =+ x x (d) x x ")1(3 2 =+
2.2 $ORMIN QUADRATIC EQUATIONS $ROM ROOTS PEM#ENTUKAN PERSAMAAN KUADRATIK DARIPADA PUNCA$PUNCANYA
1. :f !))(( = x x & then !! == xor x and the roots are %nd . ?i$% !))(( = x x , $a a !! == x%t%u x d%n punc%-punc%n)% i%#%* d%n .
2. ;n the other hand& if given dan as the roots of a quadratic equation& then&Se'%#i$n)%, =i$% di'eri d%n i%#%* punc%-punc% pers%&%%n $u%dr%ti$, &%$%,
Met+o- 1 +te s to for$ a quadratic equation are& ! %ng$%*-#%ng$%* &e&'entu$ pers%&%%n $u%dr%ti$ i%#%*( !))(( = x x
!)(2 =++ x x where& + is the roduct of roots ( POR ) !i%#%* H%si# "%&'%* Punc% @!H"P(A is the su$ of roots( SOR ) !i%#%* H%si# D%r%' Punc% @!HDP(A
Met+o- 2 +te s to for$ a quadratic equation are ! %ng$%*-#%ng$%* &e&'entu$ pers%&%%n $u%dr%ti$ d%rip%d% i%#%*5(
(i)
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E'er%!&e 2.2.1
/.
Note 7:f given the equation as !2 =++ c'x%x & then the x coefficient need to be e ressed into the value of 1.! Not% ?i$% di'eri !2 =++ c'x%x , per#u diung$%p$%n d%*u#u pe$%#i x2 sup%)% &en=%di s%tu, i%itu(
+a
c &
a
, & ' =++
#hen& !&%$%(, su$ of roots&%'
SB/ =+= ( H%si# t%&'%* Punc%,%
' H"P =+= (
roduct of roots& %c PB/ == !H%si# d%r%' Punc%,
%
c HDP == (
E'am le
:f %nd are the roots of the quadratic equation !523 2 = x x , for$ the quadratic equation which hasthe roots 22 %nd +olution7%iven the quadratic equation& !523 2 = x x#hen& 52&3 === c%nd '%#he roots of the quadratic equation are dan
#hen&3
2
3
2
%
'SB/ ===+=
3
9
%
c PB/ ===
#he new roots are 22 dan
;
3:
3
2
SB/
=
+==
+=+=
3
1!
/
4
3
52
2
22)(22
3
9-
2 (!
PB/
/
252
)2)(2(
==
==
#he new quadratic equation for$ed
4
;
29 x
;
3:2 x
4 PB/ x (SB/! 2 x
=+
=+
3 & ' -.& '/ * +
1. 'or$ the quadratic equation fro$ the given roots as shown in the table7
Root&Q a-rat!% E, at!on
$a%tor!&at!on Met+o- SOR4POR Met+o-
(a)
(b)
2 and >3
2 and 5
Root $a%tor!&at!on Met+o- SOR4POR Met+o-
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(c)43 and 6
(d)91
-%nd 32
(e).$ 2
and 4
2. 'ind the value of & and $ for each of the following quadratic equations with the roots given.( ari nilai & dan nilai $ bagi setia ersa$aan uadrati dengan uncanya diberi).
(a) !3 2 =++ $ &x x with roots 5 dan3
1. (b) !2 2 =+ $ &x x with roots 1 dan
2
1 .
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1!.(c) x$ x )1(/2 2 = with roots 3 and2
&. (d) x:3 x 2 =+ with roots & and $ .
3. 'ind the value of p for each of the following quadratic equations.( ari nilai p bagi setia ersa$aan uadrati beri ut ).
(a) ;ne of the roots of the quadratic equation3x2 7 px + 9: 4 is twice of the other root.
(+atu dari ada unca ersa$aan !5423 =+ px x ialah dua ali unc yang satu lagi).
(b) ;ne of the roots of the quadratic equation x2 px + 12 4 is thrice of the other root.!S%tu d%rip%d% punc% pers%&%%n !12
2 =+ px x i%#%* tig% $%#i punc )%ng s%tu #%gi(.
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t%ngen
11.(c) ;ne of the roots of the quadratic equation x2 px+ 8 4 is square root of the other root.(+atu dari ada unca ersa$aan x2 px + 8 4 ialah unca uasadua unca yang satu lagi).
(d) ;ne of the roots of the quadratic equation x2 6x 2px 7 2 is square of the other root.(+atu dari ada unca ersa$aan x2 6x 2px 7 2 ialahuasa dua unca yang satu lagi).
2. CONDITION $OR T5PES O$ ROOTS O$ QUADRATIC EQUATIONS(S5ARAT UNTU6 7ENIS PUNCA PERSAMAAN 6UADRATI6)
1. #y es of roots of quadratic equations %x2
+ 'x + c 4 de end to the value of '2
- :%c which derived fro$
%%c''
x2
42 =
2. (?enis unca ersa$aan uadrati !2 =++ c'x%x bergantung e ada nilai %c' 42 yang wu@ud dari ada ru$us uadrati &
%
%c'' x
2
42
= ).
#he foolowing table shows the ty es of roots of quadratic equations. (?adual di bawah $enun@u an sifat unca ersa$aan uadrati ).
!42 > %c' !42 = %c' !42
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E'er%!&e 2. .1
t%ngent #ine
12.
#wo distinctive roots !Du% punc% n)%t%(
E'am le
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(a) !522 = x x
Q a-rat!% E, at!on :al e of %c' 42 Con-!t!on of t+e Root&
(b) !/62 =+ x x
(c) 632 = x x
(d) 3-3 2 += x x
(e) 3)41( = x x
2. #he following quadratic equations have two equal roots& deter$ine the ossible value of p. ! Pers%&%%n $u%dr%ti$ 'eri$ut &e&pun)%i du% punc% )%ng s%&%, c%ri ni#%i )%ng &ung$in '%gi p.(
(a) !22 =++ p x px (b) !2-2 =+ x px
3.
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4. 'ind the range of $ values if the following quadratic equations do not have any distinctive root.( ari @ulat nilai $ @i a ersa$aan uadrati beri ut tiada unca).
(a) !32 =+ $ x x (b) !342 2 =++ $ x x
5. C ress a relationshi between p and F if the following quadratic equations have two equal roots. (#erbit an suatu er aitan antara p dengan F @i a ersa$aan uadrati beri ut $e$ unyai dua unca yang sa$a).
(a) !4/2 =+ pFx px (b) !/52 =++ pFx px
6. C ress * in ter$s of $ if the quadratic equation!55 2 =++ $ *x$x have two distinctive different
roots.
". C ress p in ter$s of F if the quadraticequations 4F p x6 x 2 =++ do not have any root.
14.
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1. +olve the quadratic equation2x!x 7 :( !1 7x(!x+2(. %ive your answer correct to four significant figures.
D3 $ar sESPM2443GP%per 1
2. #he quadratic equation x!x + 1( px 7 : has twodistinct roots. 'ind the range of values of p.
D3 $ar sE SPM2443GP%per 1
3. 'or$ the quadratic equation which has the roots3 and F & in the for$ %x2 + 'x + c 4, where a&
b and c are constant.D2 $ar sE
SPM244:GP%per 1
4. #he straight line ) 9x 7 1 does not intersect thecurve ) 2x 2 + x + p .'ind the range of values of p.
D3 $ar sE SPM2449GP%per 1
Pa&t 5ear SPM e&t!on& 15.
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16. 1".
5. +olve the quadratic equation x!2x 7 9( 2x 7 1.%ive your answer correct to three deci$al laces.
D3 $ar sESPM2449GP%per 1
6. B quadratic equation 2 4 2 has two equal roots. 'ind the ossible values of p. D3 $ar sE
SPM2446GP%per 1
". (a) +olve the following quadratic equation3 2 5 2 !.
(b) #he quadratic equation *x2 + $x + 3 4, where* and $ are constants& has two equal roots.C ress * in ter$s of $. D4 $ar sE
SPM244 GP%per 1
-. :t is given that 1 is one of the roots of thequadratic equation x2 7 :x 7 p 4.
'ind the value of p. D3 $ar sE SPM2448GP%per 1
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