STO ELEMENTARNIH Zadataka Rjesenja

1

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

**** MLADEN SRAGA ****

POTPUNO RIJEŠENI ZADACI PRIRUČNIK ZA SAMOSTALNO UČENJE

100 ELEMENTARNIH ZADATAKA koje bi svatko trebao znati riješiti

I

ALGEBARSKI IZRAZI

a b a b a b a ab b

a b b a

a b a b a b a ab b

a b b a

a b a b

a b a b a b

a b a a b ab b

a b a a b ab b

a b a b a ab b

a b

+ = + ⋅ + = + +

+ = +

− = − ⋅ − = − +

− = −

− − = +

− ⋅ + = −

+ = + + +

− = − + −

− = − ⋅ + +

+

b g b g b gb g b g

b g b g b gb g b gb g b gb g b gb gb g

b g d i

2 2 2

2 2

2 2 2

2 2

2 2

2 2

3 3 2 2 3

3 3 2 2 3

3 3 2 2

3

2

2

3 3

3 3

3 2 2

2 2 2 2 2 2 2

2

2 2

= + ⋅ − +

+ + = + + + + +

RSTUVW

RSTUVW

+ + =+ =

⋅ == + ⋅ +

+ =

⋅ = ⋅+ + = = + + + =

a b a ab b

a b c a b c ab ac bc

x px qm n p

m n qx m x n

m n b

m n a cax bx c ax mx nx c

b g d ib g

a f a f

. . .

POTENCIJE

a a aa a a

aa

a a a

a bc a b c

a a

ab

ab

ab

ba

ba

aa a

aa

aa

n m n m

n m n m

n

mn m n m

n n n n

n m n m

n n

n

n n n

n

nn

⋅ =

=

= =

=

=

FHGIKJ =

FHGIKJ = FHG

IKJ =

=

=

=

=

+

−

−

⋅

−

−

−

:

:

b gd i

0

1

1

1

1

1

α

TEL. 01-4578-431

© M.I.M.-SRAGA 1999./2010.

2

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

Autor: MLADEN SRAGA Grafička obrada MLADEN SRAGA Izdavač:

M.I.M.-Sraga d.o.o. Centar za poduku www.mim-sraga.com

Procijena i ocijena znanjaako se spremate :

Za polaganjeZa polaganje OSNOVNE razine VIŠE razine

0 50 0 0 50 0

51 75 1 51 81 176 85 2 82 86 286 91 3 87 90 392 95 4 91 94 4

96 100 5 95 100 5

Moj kome

− = − =

− = − =− = − =− = − =− = − =− = − =

hitno tražite instruktora

trebali bi ozbiljno poćeti vježbati

Vi znate nešto ali ima tu puno rupa

ntar:ako ste dobili ocjenu 0 -ako ste dobili ocjenu 1 - ako ste dobili ocjenu 2 - ako ste dobili ocjenu dobro možete krenut na pripreme

vi ste spremni za OZBILJNIJE zadatke

odlično- vama osnovni zadaci na mature

3 - ako ste dobili ocjenu 4 - ako ste dobili ocjenu 5 - nebi trebali predstavljat problem

Prvo riješite zadatke pa tek onda gledajte u rješenja jer su tamo kompletno riješeni....

Potpunu garanciju na zadatke i rješenja daje: centar za dopisnu poduku M.I.M.-SRAGA -dakle sve što vam se čini nejasno krivo ili sumnjivo - zovite 01-4578-431 ili 01-4579-130 i tražite dodatne upute i objašnjenja ...

Zadatci su u potpunosti naš autorski rad , ako se neki zadatak već pojavio u nekoj drugoj zbirci ...nije namjerno ponovljen s naše strane.

Centar za poduku M.I.M.-Sraga zadržava sva prava na komercijalno iskorištavanje ovih zadataka. Svako kopiranje umnažanje za vlastite potrebe i potrebe poduke i nastave je dozvoljeno i dobrodošlo. Cijena ovih zadataka je 0 kn tj. besplatni su ...i tako treba i ostati... Sve kritike i dobre i loše rado ćemo saslušati ... javite nam se na tel. 01-4578-431 ili e-mailom : mim-sraga@zg.htnet.hr

100 Elementarnih zadataka

3 M.I.M.-SRAGA 1999./ 2010.©

Kompletna rješenja s postupkom

( )

( )( )( )( ) ( )

2 3 2 3 5

2 4 6 2 4 6 12

2 2 2 6 5 32 2 5 315 22 3 23 3 2 62

4 2 4 2 2

7 2 3 7 2 3 2

7 8 9 7 8 9 6

88 3 8 3 5

3

32 2 3 6

543 3 4 5 60

26 52 2 2 6

1)2)

3)4) :5) : :6) :

7 ) :

8)

9)

10)

+

+ +

⋅ + ⋅ + ⋅+ + ⋅

−

− −

+ −

−

⋅

⋅ ⋅

⋅ ⋅

⋅ = =

⋅ ⋅ = =

⋅ ⋅ = = =

= =

= =

⋅ = =

= = =

= =

= =

⋅ =

x x x xx x x x x

x x x x x xx x x xx x x x xx x x x xx x x x xx

x x x

x x x

x x x

( )( ) ( )( ) ( )

( ) ( )

( ) ( ) ( ) ( )( )

2 2 5 24 10 24 10 34

35 52 4 2 5 3 4 5 30 20 30 20 10

3 32 3 3 2 3 2 3 3 6

4 4 4 42 3 2 3 8 124

3 32 3 3 22 3 2 2 3 2 2 2 6 9 23

11) : : :

12) 2 2 8 8

2 2 1613)3 3 81

2 2 814) : 2 : 2 : 43 3 27

⋅ +

⋅ ⋅ ⋅ −

⋅

⋅ = ⋅ = =

= = = =

= ⋅ ⋅ = ⋅ ⋅ =

⎛ ⎞ = ⋅ ⋅ =⎜ ⎟⎝ ⎠

⎛ ⎞ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

x x x x x

x x x x x x x x

ab a b a b a b

x y x y x y

a b ab a b a b a b a( )

( )( ) ( )

( )( )

4

6 9 2 4 4 5 4 5

2 4 2 4

1

2 3 2 3 2 3 2 3 5

3 2 3 2 3 5

3 23 2 2 222 2 2 6

3 2 23 3 4 93 3 4

8 1 4 2 1 227 4 27 4 27

15)

16)

17 ) : :

−

− +

+

⋅ =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= ⋅ = ⋅ =

⋅ ⋅ ⋅ ⋅

⎛ ⎞=⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ = ⋅ = ⋅ = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ = ⋅ =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

b

a b a a b b a ba b a b

x yy x

x y y y y y y yy x x x x x x x

a ya y y y ay a a yy a a

2 4 6 2 8

6 8 9 6 4

2 8 8 8

9 2 2 9 2 11

1 2 3 2 3

10 0 0 1

10 1

:

2 1 1 3 3 2 3 3 3 2 518) 9 8 13 3 2 2 1 1 2 2 2 2

1 319) 27 2 1 1 1 2 2 3 63 1

1 120) 22 5

+

− − −

−

−

⋅ = ⋅ ⋅ =

⋅= = =

⋅ ⋅

+⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − = + − = + − = + = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅⎜ ⎟ ⎜⎝ ⎠ ⎝

y y a y aa a y a y

y a a ay y y y y

x y

y1

33

1 1 5 1 1 5 1 52 12 2 1 2 2 2 8 32

−− ⋅ ⋅

⋅ = ⋅ ⋅ ⋅ ⋅ = =⎟ ⋅ ⋅⎠

100 Elementarnih zadataka

4 M.I.M.-SRAGA 1999./ 2010.©

( )

( )

12

12

12

2 1 12 2 3 23 3 3

34

3 1 3434 4

122

14 444

12

1 4444 44

4

1 12 2

)

1 1)

)

1 1 1)

) 36 36 6 6

) 16 16 2 2

25 25 25 5)49 49 749

16 16 2 2 2)81 81 3 3 3

64 49 49)49 64

−

⋅

−

−

=

= =

= = =

= = =

= = =

= = =

⎛ ⎞ = = =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x x

xxx

x x x x

xxxx

21 22 23 24 25 26

27 28

29

333 3 32 2 32 3222 2 2

2 3

2

3 33 3 33

4 44 4 44

5 2 5 3 2 3 5 55

6 5

49 764 864

9 16 4 4 4 4 4 64)16 9 3 3 3 3 3 27

1) 10 10 10 10 10 10

) 3 9 3 9 27 3 3

) 2 8 2 8 16 2 2

)

) :

− ⋅

= =

⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = = = = =⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⋅ = ⋅ = =

⋅ = ⋅ = = =

⋅ = ⋅ = = =

⋅ = ⋅ = =x x x x x x

x

30

3 32

33 34

2 36 5 2 6 5 2 6 3 3 26 : ⋅−= = = = = =x x x x x x x x 35

5

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

( )

( ) ( ) ( )

( ) ( ) ( )

2 2 2

2 2 2 2 2 2 2

2 2 2 2 2 2 2

2 2 2 23 3 3 2 3

2

2

2 2

2 2

36) 3 2 3 3

37 ) 2 3 2 2 2 3 3 2 12 3

38) 3 4 3 2 3 4 4 3 24 4

1 2 1 1 2 2 1 239) 22 3 2 2 3 3 32

6 9

4 12 9

9 24 16

x x x

x y x x y y x xy y

x y x x y y x xy y

x y x x y y x xy

x x

x xy y

x xy y

+ = + ⋅ ⋅ + =

+ = + ⋅ ⋅ + = + + =

− = − ⋅ ⋅ + = − + =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = + ⋅ ⋅ + = + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

+ +

+ +

− +

( )

( ) ( )

( )( ) ( ) ( )

( ) ( )( ) ( )

2 23 32

2 2 2 2 22 23 4 3 3 4 4 3 3 4 32 2

2

2 2 2

22

2 6

6 3 4 8

2 2

23

2 2 32 3 2 2 3 3 2 340) 23 4 3 3 4 4 3 43 4

1 141) 5 65 6 5 2 5 6 6

42) 2 3 1 2 3 1

1 2 44 3 9

4 99 16

125 60 36

y

x y x x y y x x y x

x yx y x x y y

a y a y

x xy y

x x y y

x xy y−

=

⋅ ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − ⋅ ⋅ + = − ⋅ ⋅ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠

=

− = = =− − ⋅ ⋅ +

− − = − ⋅ + = −

+ +

− +

− +

( ) ( ) ( )( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

2 2 2 2

2 2 2 2

2 2 2 22 22 3 2 3 2 3 2 32 2

22 3 4 5 6 2 3 4 5 6 2 3 4

2 2

2

4 6

2 3 1 2 2 2 3 3

43) 2 5 2 5 2 5 2 25

2 4 2 4 2 4 2 444)3 5 3 5 3 5 3 5

45) 3 4 3 4 3 4

4 12 9

4 25

4 169 25

a y a a y y

x x x x

x y x y x y x y

x y z a b x y z a b x y z

a ay y

x

x y

⋅ + = ⋅ + ⋅ ⋅ + =

− ⋅ + = − = − =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− ⋅ + = − = − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

− ⋅ − = −

+ +

−

−

( )( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

25 6

2 2 2 2 22 2 3 4 2 5 6

3 3 2 2 3 3 2

2 2 1 2 1 3 3 2 2 3

3 3 2 2

4 6 8 10 12

3 2

3 2

3 4

46) 3 3 3 3 3 3 9 3 9 27

47) 2 2 2 2 2 2 3 2 3 2 2

48) 2 3 2 3 2 3 3 2 3 3

9 16

9 27 27

6 12 8

a b

x y z a b

x x x x x x x

x x x x x x x x x

y z y y z y z

x y z a b

x x x

x x x+

=

= − =

+ = + ⋅ ⋅ + ⋅ ⋅ + = + + ⋅ ⋅ + =

+ ⋅ + = + ⋅ + = + = + = + ⋅ ⋅ + ⋅ ⋅ + =

− = − ⋅ ⋅ + ⋅ ⋅ −

−

+ + +

+ + +

( )3 3 3 2 2 2 2 3 3

3 3 2 2 3 3 32 3

3 3

32 3 2

3 2 2 3

2 3

2 9 2 6 3 3

1 1 1 1 1 1 1 1 1 1 1 1 1 149) 3 3 3 32 3 2 2 3 2 3 3 4 3 2 92 3

2 3 250)3 2 3

8 36 54 27

1 1 1 18 4 6 27

z y y z y z z

x x x x x x x

x y x

y y z yz z

x x x

= − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − =

=

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = + ⋅ ⋅ + ⋅ ⋅ + = + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

=

⎛ ⎞ ⎛− =⎜ ⎟⎝ ⎠

− + −

+ + +

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

3 2 2 32 3 2 3 3

3 2 2 33 2 2 32 2 3 2 3 33 2 2 3

3 3 2 2 32 3 4 2 3 2 3 4 2 3 4 4

3 3 23 2 3 2

6 4 3 2 6 9

2 3 2 3 33 33 2 3 2 2

2 2 3 3 33 223 3 2 2

51) 3 2 3 3 3 2 3 3 2 2

3 3 2 9

8 9 27227 2 8

x y x y y

x x y x y y

x y z x y x y z x y z z

x y x y

x x y x y y

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− ⋅ ⋅ + ⋅ ⋅ − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= − ⋅ ⋅ + ⋅ ⋅ − =

=

− = − ⋅ ⋅ + ⋅ ⋅ − =

= − ⋅ ⋅ ⋅ ⋅

− + −

( ) ( ) ( )

( ) ( ) ( )

2 2 33 4 2 3 4 3 4

3 3 3 2 2 3 3 2 2 3

3 2 2 3 3 2 2 3

6 9 4 6 2 3 8 12

2 3

9 4 2

52) 3 3 3 3

3 3 3 3

27 54 36 8

6 2

z x y z z

x y x y x x y xy y x x y xy y

x x y xy y x x y xy y

x y x y x y z z

x y y

⋅ + ⋅ ⋅ − =

=

+ − − = + + + − − + − =

= + + + − + − + =

− + −

+

6

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

7

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

100 Elementarnih zadataka

8 M.I.M.-SRAGA 1999./ 2010.©

9

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

10

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

10

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

11

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

12

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

Ovo su uglavnom zadaci iz PRVOG razreda gimnazije oni će rijetko doći u ovom obliku, već su sastavni dio težih ( složenijih ) zadataka ali ako ih ne znate riješiti nećete znati riješiti ni zadatke kojih su oni dio… Naš prijedlog provježbajte ih koristit će Vam !!!!!! Moguće da je jedan zadatak krivo riješen ako ga pronađete javite nam: mim-sraga@zg.htnet.hr Ako ne vidite dobro tekst od 15 do 20 strane zum namjestite na 150%

13

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

STO ELEMENTARNIH ZADATAKA KOJE BI SVI TREBALI ZNATI RIJEŠITI

ako imaju namjeru položiti maturu ! Koristeći ove formule trebali bi riješiti zadane zadatke:

( )

( )( )( )

0

1

1

1

1

1

POTENCIJE

:

:

1

1

1

1 1

1 1

n m n m

n m n m

nn m n m

m

n n n n

mn n m

zmn n m z

n n

n

n n n

n

nn

nn

nn

n

mn mn

mn

m n mn

a a aa a a

a a a aa

abc a b c

a a

a a

a ab b

a b bb a a

aa a

aa

aa

a a

aaa

a a

aaa

+

−

−

⋅

⋅ ⋅

−

−

−

−

−

⋅ =

=

= =

=

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=

=

=

=

=

= =

=

= =

( )

( )( )( )( ) ( )

( )( ) ( )

( )

( )

2 3

2 4 6

2 523 2

4 2

7 2 3

7 8 9

8

3

32

543

26 52 2

35 52 4

32

42 3

322 3 2

1

2 3

1)2)

3)4) :5) : :6) :

7 )

8)

9)

10)

11) :

12) 2

213)3

214) : 23

15)

16)

x xx x x

x x xx xx x xx x xxx

x

x

x x

x x

ab

x y

a b ab

xy

x yy x

−

−

⋅ =

⋅ ⋅ =

⋅ ⋅ =

=

=

⋅ =

=

=

=

⋅ =

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞=⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

3 222 2

3 3 4

1 2 3

10 0 0 1

1 10 1 3

17 ) :

2 1 118)3 3 2

119) 27 23

1 120) 2 22 5

− − −

−

−− −

⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⋅ ⋅ ⋅ ⋅ =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ ⋅ ⋅ ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

a y yy a a

x y

y

14

100 Elementarnih zadataka

M.I.M.-©

12

12

23

34

12

14

12

14

12

32

3 3

4 4

5 2 5 3

6 5 6 2

21)

22)

23)

24)

25) 36

26) 16

2527 )49

1628)81

6429)49

930)16

31) 10 10

32) 3 9

33) 2 8

34)

35) :

x

x

x

x

x x

x x

−

−

−

−

=

=

=

=

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

⋅ =

⋅ =

⋅ =

⋅ =

=

( )

( )

( )( )

0

1

1

1

1

1

POTENCIJE

KORIJENI

:

:

1

1

1

1 1

1 1

n m n m

n m n m

nn m n m

m

n n n n

mn n m

zmn n m z

n n

n

n n n

n

nn

nn

nn

n

mn mn

mn

m n mn

n n n

n n

a a aa a a

a a a aa

abc a b c

a a

a a

a ab b

a b bb a a

aa a

aa

aa

a a

aaa

a a

aaa

a b a b

aa b

+

−

−

⋅

⋅ ⋅

−

−

−

−

−

⋅ =

=

= =

=

=

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=

=

=

=

=

= =

=

= =

⋅ = ⋅

⋅ = n

b

SRAGA 1999./ 2010.

15

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

Koristeći formule koje smo upravo obradili i ove nove koje vam dajem ovdje riješite sljedeće zadatke: ( )

( )

( )

( )

( )

( ) ( )

( ) (

( )

( ) ( )

( )

2

2

2

23

23 4

2

2

2 3 2 3

2 3 4 5 6 2 3 4 5 6

3

2

3

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )( ) ( )

( ) ( )( )( )

( ) ( )( ) ( )

( )

2 2 2

2 2

2 2 2

2 2

2 2

2 2

3 3 2 2 3

3 3 2 2 3

3 3 2 2

3 3 2 2

2 2 2 2

ALGEBARSKI IZRAZI

2

2

3 3

3 3

2 2

a b a b a b a ab b

a b b a

a b a b a b a ab b

a b b a

a b a b

a b a b a b

a b a a b ab b

a b a a b ab b

a b a b a ab b

a b a b a ab b

a b c a b c ab a

+ = + ⋅ + = + +

+ = +

− = − ⋅ − = − +

− = −

− − = +

− ⋅ + = −

+ = + + +

− = − + −

− = − ⋅ + +

+ = + ⋅ − +

+ + = + + + + 2c bc+

)

) 3

) 2 3

) 3 4

1 2

45

45) 3 4 3 4

46) 3

47 ) 2 2

48) 2 3

49

+ =

+ =

− =

⎛ ⎞

⎞+ =⎟⎠

− ⋅ − =

+ =

+ ⋅ + =

− =

x

x y

x y

y

x y z a b x y z a b

x

x x

y z

36 37

38

39)

2 3

2 340)3 4

41) 5 6

42) 2 3

43) 2 5 2 5

2 4 244)3 5 3

−

+ =⎜ ⎟⎝ ⎠

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =

− − =

− ⋅ + =

⎛ ⎞ ⎛− ⋅⎜ ⎟ ⎜⎝ ⎠ ⎝

x y

x y

x y

a y

x x

x y x

( )

( ) ( )

3

32 3

32 3 4

3 3

1 1)2 3

2 3)3 2

) 3 2

)

⎛ ⎞+ =⎜ ⎟⎝ ⎠

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =

+ − − =

x

x y

x y z

x y x y

50 51 52

16

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

P

( ) ( )( ) ( )

3 3 2 2

3 3 2 2

ALGEBARSKI IZRAZI

a b a b a ab b

a b a b a ab b

− = − ⋅ + +

+ = + ⋅ − +

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( )

( ) ( )

( ) ( )

3 6

3 3 23 23 6 3 3 3 2 2 2 2 2

333 3 2

2

2 2 2 4

3 3 2 2Prepoznaj:

rimjer:27 8 ?

ešenje:

27 8 3 2 3 2 3 2

x y

x y x y x y y x x y y

a x b y

a x b y

x y x xy y

a b a b a a b b

− =

− = − = − = − ⋅ + ⋅ + =

↓ ↓

↓ ↓

= =

= =

= − ⋅ + +

− = − ⋅ + ⋅ +

rj

x 3 3 2 2

3 2

3 2

3 2 9 6 4

Koristi gornji primjer i ove formule riješi sljedeće zadatke: Rastavi na faktor

3

3

6

6 6

3

9 6 3

e:

) 8

) 8 27

) 125

)

1) 127

) 64 125

x

y

x

x y

y

x y z

− =

− =

+ =

− =

+ =

+ =

53

54

55

56

57

58

17

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

( )2 2 2 2 2 2 2a b c a b c ab ac bc+ + = + + + + +Koristeći se formulom riješi sljedeće zadatke: ( )

( )

( )

2

2

2

) 2 3

3

x y

x y

+ + =

+ − =

59

60) 2 3 4

) 3

x y z+ + =

61

( ) ( )

( ) ( )

( ) ( )

2

2 2

4 6

4 4

2 4 8

2 2

22

2 2Koristeći se formulom:

rastavi na faktore:

) x 16

) 4 36

1 25)4 49

)

) 4

) 2

) 36 16

y x

x y

x y

x y z

x y x y

x y x y

a b a b a b

− =

− =

− =

− =

− =

+ − − =

+ − − =

− = − ⋅ +

62

63

64

65

66

67

68

18

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

( ) ( )

2 2

2 2

6 3 4 8

4 6 2 3

2 4 8

2 2 4

2 22 2 2 2 Koristeći se formulama:

slijedeće zadatke zapiši u obliku kvadrata binoma:

) 9 6

) 2

) 2

1) 3 94

2 2

x y y

x xy y

x x y y

x y x y

x xy y

a ab b

a ab b a b a ab b a b

+ + =

− + =

− + =

+ + =

− + − =

− + =

+ + = + − + = − 69

70) 4 20 25

71) 9 12 4

72 73 74

( ) ( )2

2 2

2

2

2

...

Kvadratni trinom

Koristeći se gornjim formulama rastavi na faktore slijedeće kvadratne trinome:

75) x 2 15

76) 3 2

77 )

ax bx c ax mx nx c

m n px px q x m x n

m n q

m n b

m n a c

x

x x

x x

+ + = = + + + =

+ =+ + = = + ⋅ +

⋅ =

+ =

⋅ = ⋅

⎧ ⎫⎨ ⎬⎩ ⎭

⎧ ⎫⎨ ⎬⎩ ⎭

+ − =

+ − =

− 12− =

19

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

Skra

( )

( )

2 2

2 2

4 4

2 2

2

2 2

2

4 4

3 3

2 2

3

Pogledaj u formule za ove brojnike u 81) i 82)

ti razlomke koristeći sve što smo izvježbali do sada:

4 6)6 9

)2

)2

)

)

)3 3

)

x yx y

x yx xy y

a ba ab b

x yy x

y xx y

x yx y

x

−=

−

−=

+ +

−=

− +

−=

−

+=

−

−=

−

( )

78

79

80

81

82

83

3

3

2

2

2 2

2 2

3 18 27)2 18

2 3)3 6 3

yx y

x xx

x xy yx xy y

+=

+

+ +=

−

− +=

+ +

84

85

86

20

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

( )

( )

2

2

2

2 2 4 4

2 2 2

3 2)2 3 4

1 2)2 2

3 2)1 1 1

1 1)3 3

1)

)

x x x

x x

xx x x

y y

yx y x y

y yx y y x

+ −− + =

− =+ −

−+ − =

− + −

− =− −

− =+ −

− =− −

87 87

2 3 88 89 90

91

Iz

( )

2 2 2 2

3 3 2

4 4 6 3 3 6

3 3 4 2 2

4

2

računaj:

4 4 4)

2)

1 2 1)1 1 9 1

x y x xy yx y y x

x y x x y yx y x x y

xx x x

− + +⋅ =

− +

− + +⋅ =

+ +

−⎛ ⎞+ ⋅ =⎜ ⎟+ − −⎝ ⎠

92

93

94

21

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©

( )( )

2 2

2 2

2 2 2

2 22

3 2

2 2

2 3

2

3 18 27 995) :6 24 4 4

2) :

1 1 3 2 2)1 1 1 6 9

111)

111

111 2)11

1 2

00)4 4

x y xy y xx y y x x

x y y yx yxz yz zx y z

x x x xx x x x x

y

y

xxxx

x y x yx y x y

x y xyxy x y y

+ + −=

− + +

+ − +=

− −− −

− + +⎛ ⎞− + ⋅ =⎜ ⎟− + − −⎝ ⎠

+−

=−

+

+−

− =+

+−

− +−

+ −=

−− + −

96

97

98

99

1

22

100 Elementarnih zadataka

M.I.M.-SRAGA 1999./ 2010.©