Quadratic Eq n 12

8/12/2019 Quadratic Eq n 12

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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


13/17

(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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Quadratic Eq n 12