RST UVW

( )
2 2 2
36) x+ 3 = x + 2⋅x⋅ 3 + 3 = x
2
+ 6x+
9
100 Elementarnih zadataka
( ) ( ) ( )
2 2 2 2 2 2 2
37 ) 2x+ 3y = 2x + 2⋅2x⋅ 3y+ 3y = 2 x + 12xy+ 3 y = 4x + 12xy+
9y
( ) ( ) ( )
2 2 2 2 2 2 2
2 2
38) 3x− 4y = 3x −2⋅3x⋅ 4y+ 4y = 3 x − 24xy+ 4 y = 9x − 24xy+
16y
2 2
2 2 2 2
3 3 3 2 3
2
⎛1 2 ⎞ ⎛1 ⎞ 1 2 ⎛2 ⎞ 1 2 2
39) ⎜ x+ y ⎟ = ⎜ x⎟ + 2⋅ x⋅ y + ⎜ y ⎟ = x + xy +
⎝2 3 ⎠ ⎝2 ⎠ 2 3 ⎝3 ⎠ 2 3 3
2
2
( y )
1 2 4
= x + xy + y
4 3 9
2
3 2 3 6
2 2 2 2 2
2 2
3 4 3 3 4 4 3 223 ⋅ ⋅ 3 4 3
2 ( ) 2 ( )
40)
⎛2 3 ⎞ ⎛2 ⎞ 2 3 ⎛3 ⎞ 2 3
⎜ x − y ⎟ = ⎜ x ⎟ −2⋅ x ⋅ y + ⎜ y ⎟ = x − ⋅x ⋅ y +
⎝3 4 ⎠ ⎝3 ⎠ 3 4 ⎝4 ⎠ 3 3⋅4
4
x =
4 6 3 4 9 8
= x − x y + y
9 16
41)
−2
1 1
1
( 5x− 6y)
= = =
2 2 2 2 2
25x − 60xy+
36y
( 5x−6y) ( 5x) −2⋅5x⋅ 6y+
( 6y)
2
2
( − a− y) = ( − ⋅ ( a+ y)
) = ( − )
42) 2 3 1 2 3 1
( ) ( ) ( )
2 2 2 2
43) 2x−5 ⋅ 2x+ 5 = 2x − 5 = 2 x − 25 =
( )
( 2a 3y) 1 ( 2a) 2 2a 3y ( 3y)
2 2 2 2
⋅ + = ⋅ + ⋅ ⋅ + = 4a + 12ay+
9y
2
4 25
x
−
2 2
44)
2 2 2 2
⎛2 2 2
2 4 3⎞ ⎛2 2 4 3⎞ ⎛2 2⎞ ⎛4 3⎞
2 2 4 3 4 16
⎜ x − y ⎟⋅ ⎜ x + y ⎟ = ⎜ x ⎟ − ⎜ y ⎟ =
2 ( x ) −
2 ( y ) = x − y
⎝3 5 ⎠ ⎝3 5 ⎠ ⎝3 ⎠ ⎝5 ⎠ 3 5 9 25
4 6
2
2
2 3 4 5 6 2 3 4 5 6 2 3 4
5 6
( x y z − a b ) ⋅( x y z − a b ) = ( x y z ) −( ab ) =
2 2 3 4 2 5 6
3 ( x ) ( y ) ( z ) 4 ( a ) ( b )
45) 3 4 3 4 3 4
( )
3 3 2 2 3 3 2
+
( ) ( ) ( ) ( ) ( ) ( )
( y z) ( y) ( y) z y ( z)
2 2 2 2 2
46) x+ 3 = x + 3⋅x ⋅ 3 + 3⋅x⋅ 3 + 3 = x + 9x + 3⋅9⋅ x+ 27 =
2 2 1 2 1 3 3 2 2 3
3 3 2 2
= − = 9x y z −16a b
+ 9 + 27 + 27
3 2
x x x
4 6 8 10 12
47 ) x+ 2 ⋅ x+ 2 = x+ 2 ⋅ x+ 2 = x+ 2 = x+ 2 = x + 3⋅x ⋅ 2 + 3⋅x⋅ 2 + 2 =
( ) 3 3 3 2 2 2 2 3 3
48) 2 − 3 = 2 −3⋅ 2 ⋅ 3 + 3⋅2 ⋅ 3 − 3z = 2 y −9⋅2 ⋅y ⋅ z+ 6⋅y⋅3 ⋅z − 3 z =
= 8y − 36y z+ 54yz −27z
3 2 2 3
+ 6 + 12 + 8
3 2
x x x
3 3 2 2 3 3 3
2 3
3 3
49)
⎛1 1 ⎞ ⎛1⎞ ⎛1⎞ 1 1 ⎛1 ⎞ ⎛1 ⎞ 1 1 1 1 1 1
⎜ + x⎟ = ⎜ ⎟ + 3⋅⎜ ⎟ ⋅ x+ 3⋅ ⋅ ⎜ x⎟ + ⎜ x⎟
= + 3⋅ ⋅ ⋅ x+ 3⋅ ⋅ ⋅ x +
⎝2 3 ⎠ ⎝2⎠ ⎝2⎠ 3 2 ⎝3 ⎠ ⎝3 ⎠ 2 4 3 2 9 3
x =
1 1 1 2 1 3
= + x+ x + x
8 4 6 27
50)
3
3 2 2 3
2 3 2
⎛2 3 ⎞ ⎛2
⎞ ⎛2 2⎞ 3 3 2 2 ⎛3 3⎞ ⎛3
3⎞
⎜ x − y ⎟ = x 3 x y 3 x y y
⎝3 2
⎜
⎠ 3
⎟ − ⋅⎜ 3
⎟ ⋅ + ⋅ ⋅ − =
2 3
⎜
2
⎟ ⎜
2
⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
3 2 2 3
2 3 2 2 3
2 2 2 3 3 2 3 3 3 3
= ( x
3 ) −3⋅ 2 ( x ) ⋅ y + 2⋅x ⋅
2 ( y ) −
3 ( y ) =
3 3 2 2 2
8 6 4 3 9 2 6 27 9
= x − 2x y + x y − y
27 2 8
3 3 2 2 3
2 3 4 2 3 2 3 4 2 3 4 4
( x y − z ) = ( x y ) − ⋅( x y ) ⋅ z + ⋅ x y ⋅( z ) − ( z ) =
3 3 2
3 2 3 2 3 4 2 3 4 3 4
= 3 ( x ) ( y ) −3⋅2⋅9⋅( x ) ⋅( y ) z 9x y 4 ( z ) 2 ( z )
51) 3 2 3 3 3 2 3 3 2 2
= 27x y − 54x y + 36x y z −8z
6 9 4 6 2 3 8 12
2 2 3
⋅ + ⋅ ⋅ − =
3 3 3 2 2 3 3 2 2 3
( x y) ( x y) x x y xy y ( x x y xy y )
52) + − − = + 3 + 3 + − − 3 + 3 − =
x 3x y 3xy y x 3x y 3xy y
3 2 2 3 3 2 2 3
= + + + − + − + =
6x y+
2y
2 3
©
M I M SRAGA 1999 / 2004
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