2. quadratic equations (mas)

8/12/2019 2. Quadratic Equations (Mas)

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2. QUADRATIC EQUATIONS

IMPORTANT NOTES :

(i) The general form of a quadratic equation is ax2+ bx + c = 0; a, b, c are constants and a 0.(ii) haracteristics of a quadratic equation!

(a) "n#ol#es onl$ %&' #ariable,(b) as an equal sign = * and can be expressed in the !r" ax2# bx # c $ %,

(c) The highest oer of the #ariable is 2.

2.& Rec!'nisin' Q(adratic E)(ati!ns

'-/1'&o 3uafratic 'quations (3.'.) &%& 3.'. 456

7. x2+ 2x 89 = 0 2x : 9 = 0 &o terms in x2 ( a = 0)

2. x2 = x

x 22 + = 0 Term

x

2

9.


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2.2 The ROOTs ! a )(adratic E)(ati!n *Q.E.+

&ote ! F%%T* refers to a secific #alue hich satisfies the 3.'.

'xamle ! >i#en 3.'. x2+ 2x : 9 = 0

G$ substitution, it is found that !x = 7 , 72+ 2(7) : 9 = 0

ence 7 is a root of the quadratic equation x2+ 2x : 9 = 0.Gut if x = 2, 22 + 2(2) : 9 0,

4e sa$ that 2 is NOT a root of the gi#en quadratic 'quation.

E,AMP-E E,ERCISE

C&. Hetermine if 82 is a root of the equation9x2 + 2x 8B = 0.

x $ 2/ 0*2+2 # 2*2+ 1 $ &2 1 3 1

4 %

5ence 2 is NOT a r!!t ! the 'i6en e)(ati!n.

-&. Hetermine if 9 is a root of the equation2x2: x : 7 = 0.

-2. -&. Hetermine if0 is a r!!t ! the e)(ati!n x21 2x # 0 $ %.

-0. Hetermine if 7 is a r!!t ! the e)(ati!n3x2 # 2x 1 2 $ %.

C2. I 2 is a r!!t ! the )(adratic e)(ati!n

x21 8x 1 &% $ %/ ind 8.

x $ 2/ *2+2 1 8*2+ 1 &% $ %

3 # 28 1 &% $ %

28 $ &3

8 $

-3. I 0 is a r!!t ! the e)(ati!n

x21 28x # &2 $ % / ind 8.

-9. I 2 and p are r!!ts ! the )(adratic e)(ati!n

2x2# 0x # 8 $ %/ ind the 6a(e ! 8 and p.

-;. I & are r!!ts ! the )(adratic

e)(ati!n px21 3x # 0p 1 < $ %/ ind p.

2 3uadratic 'quations 2


3/13

Ho $ou Cno that


I the PRODUCT ! t=!

n("bers is >er!/ then either !ne

!r b!th the n("bers "(st be

>er! ?

I x @ $ % /Then x$ % !r @$ %

!r x$ @$ % *b!th are >er!es+

Exa"pe : I*x 1 2+ *x # 0+$ % /Then x 1 2$ % !r x # 0 $ %

i.e. x$ 2 !r x$ 0 .

2 and 0 are caed the roots ! thee)(ati!n *x2+*x#0+ $ %.


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2.9.7 To ol#e 3uadratic 'quations ! ax2+ bx + c = 0I. B@ act!risati!n

8 This method can onl$ be used if the quadratic exression can be factorised comletel$.

E,AMP-E E,ERCISE

C&. S!6e the )(adratic e)(ati!n x

2

# 9x # ; $ %.Answer! x2 + x + A = 0 (x + 2) (x + 9) = 0 x + 2 = 0 or x + 9 = 0 x = 82 or x = 89

-&. S!6e x2

1 3x 1 9 $ %.Jawapan!

Ans : 1 & / 9

C2. S!6e the )(adratic e)(ati!n 2x *x 1 &+ $ ;.

Ans! 2x (x : 7) = A 2x2: x : A = 0 (2x + 9) (x : 2) = 0 2x + 9 = 0 or x : 2 = 0

x =2

9 or x = 2

-2. S!6e x * & # x+ $ ;.

Ans : 1 0 / 2

-0. S!6e *x 1 0+2$ &.

Ans : 2/ 3

-3. S!6e & # 2x2 $ 9x # 3.

Ans : &/ 02

-0. S!6e *2x 1 &+2

$ 2x 1 & .

Ans : 7 / &

-3. S!6e 9x2

1 39 $ %.

Ans : 1 0 / 0

-9. Seesai8an *x 1 0+*x # 0+ $ &;.

Ans : 1 9 / 9

-;. Seesai8an 0 # x 1 3x2$ %.

Ans : 1 / &

-. S!6e x* x # 2+ $ 23.

Ans : 1 ; / 3

-


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II. B@ GC!"petin' the S)(areH

8 To exress ax2# bx# c in the !r" a*x # p+2# )

Si"pe Case : hen a $ &

E,AMP-E E,ERCISE

7. S!6e x2

# 3x 1 9 $ % b@ "eth!d !Gc!"petin' the s)(areH.

x2+


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II. Meth!d ! c!"petin' the s)(are

8 b$ exressing ax2# bx# c in the !r" a*x # p+2# )

Ka $ &/ b(t in6!6in' racti!ns =hen c!"petin' the s)(areL

E,AMP-E E,ERCISE

C0. S!6e x21 0x 1 2 $ % b@ "eth!d !

Gc!"petin' the s)(areH. Ji6e @!(r ans=erc!rrect t! 3 si'niicant i'(res.

x2: 9x : 9 = 0

22

9

2

99

22

2

+ xx = 0

2rah @ $ *x+ cuts the x8axis at TOdistinct oints.

CASE 2 b21 3ac $ %Q.E. has rea and e)(a r!!ts.

The grah @ $ *x+ touches the x8axis I The x8axis is the tangent to thecur#eJ

CASE 0 b21 3ac %Q.E. d!es n!t ha6e rea r!!ts.

>rah @ $ *x+ does not touch x8axis.

>rah is ab!6e the x8axis sincef(x) is ala$s ositi#e.

>rah is be!= the x8axis sincef(x) is ala$s negati#e.

2.9.2 Apicati!n *Reati!nship bet=een b21 3ac and the type of roots+

E,AMP-E E,ERCISE


x

a %

@$*x+

x

a %@$*x+

x

a %

@$*x+ x

a %

@$*x+

x

a %

@$*x+ x

a %

@$*x+


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C& *SPM 2%%%+

The r!!ts ! the )(adratic e)(ati!n

2x2# px # ) $ % are ; and 0.

ind

*a+ p and )/

*b+ ran'e ! 6a(es ! 8 s(ch that 2x2# px #

) $ 8 d!es n!t ha6e rea r!!ts.

Ans=er :

*a+ x $ ; / x $ 0

*x # ;+ *x 1 0+ $ %

x2# 0x &< $ %

2x2# ;x 1 0; $ %

C!"parin' : p $ ; / ) $ 0;.

*b+ 2x2# ;x 1 0; $ 8

2x2# ;x 1 0; 1 8 $ %

a $ 2/ b $ ;/ c $ 0; 8

b2

1 3ac % ;21 3*2+*0; 1 8+ %

023 # < 8 %

8 1 3%.9

-&. The r!!ts ! the )(adratic e)(ati!n

2x2# px # ) $ % are 2 and 0.

ind

*a+ p and )/

*b+ the ran'e ! 6a(es ! 8 s(ch that

2x2# px # ) $ 8 d!es n!t ha6e rea r!!ts.

-2 ind the ran'e ! 8 i the )(adratic e)(ati!n

2x21 x $ 8 has rea and distinct r!!ts.

* Ans : 8 &< +

-0. The )(adratic e)(ati!n F # 3x2$ px has

e)(a r!!ts. ind the p!ssibe 6a(es ! p.

* Ans : p $ &2 ata( &2+

-3 ind the ran'e ! p i the )(adratic

e)(ati!n 2x2# 3x # 9 # p $ % has rea r!!ts.

*Ans : p 0 +

-9. ind the ran'e ! p i the )(adratic

e)(ati!n x2# px $ 2p d!es n!t ha6e rea r!!ts.

* Ans : < p % +

-; The r!!ts ! the )(adratic e)(ati!n

2x2# < $ *8 1 0+x are rea and dierent.

Deter"ine the ran'e ! 6a(es ! 8.

* Ans : 8 9 / 8 && +

-. ind the ran'e ! 6a(es ! 8 i the

)(adratic e)(ati!n x2# 28x # 8 # ; $ % has

e)(a r!!ts.

* Ans : 8 2 / 0 +

Rein!rce"ent Exercises *SPM !r"at Q(esti!ns+

E,ERCISE E,ERCISE



12/13

-& (a) The equation x2: Ax + B = h(2x : 9) hasroots hich are equal. ?ind the #alues of h.Ii#en that K and L are roots of the equationx2: 2x + C = 0 , hile 2K and 2L are the roots ofthe equation x2+ mx + E = 0. Hetermine the

ossible #alues of C and m. I/ 7EEEJ

IAJ

( h = 87 , 82 ; C =i#en

2

pand

2

qare roots of the

equation Cx(x : 7) = m : red /T anda adalah berCadar songsang dengan latihan $ang anda buat P

2 3 d ti ' ti 79

2. quadratic equations (mas)

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