-x2 - 2x + 1 > 0
-x2 - 2x + 1 > 0
" = (-2)2 - 4 · (-1) · 1 = 4 + 4 = 8
" "
" = 2 2
"
"
"
-2 2-1
( )
2-2 2
x1 = = = 2 - 1
2·(-1) -2
"
"
"
-2 2
(-1- )
2+2 2
x2 = = = -1 - 2
2·(-1) -2
" "
x " -1 - 2; 2 - 1
" "
x " -1 - 2; 2 - 1
W (x) W (x) = x3 - 6x2 - 3x + 18
W (x) = x3 - 6x2 - 3x + 18
W (x) = x2 (x - 6) - 3 (x - 6)
W (x) = (x - 6) x2 -" "
3
W (x) = (x - 6) x - 3 x + 3
" "
W (x) = (x - 6) x - 3 x + 3
3x+4
- 2 = 0
x-1
x - 1 = 0 x = 1

3x+4
- 2 = 0
x-1
2(x-1)
3x+4
- = 0
x-1 x-1
3x+4-2x+2
= 0
x-1
x+6
= 0
x-1
x + 6 = 0
x = -6
x = -6.
W (x) = (x + 2)3 - 3 x2 - 1 - (x - 2) (x + 2) V (x) = (x + 2) x2 + 3 + 9 (x - 1)
W (x) = (x + 2)3 - 3 x2 - 1 - (x - 2) (x + 2) = x3 + 6x2 + 12x + 8 - 3x2 + 3 - x2 + 4 = x3 + 2x2 + 12x + 12
V (x) = (x + 2) x2 + 3 + 9 (x - 1) = x3 + 3x + 2x2 + 6 + 9x - 9 = x3 + 2x2 + 12x - 9
W (x) = x3 + 2x2 + 12x + 12
V (x) = x3 + 2x2 + 12x - 9
W (x) V (x)
W (x) V (x)
126km
km
62
h
s = 126km
km
V
h
t h
km
Vr = V - 62 )
h
6
t + 3 = t + 3, 1 h
60
V > 62
t > 0
Vr > 0
V · t = 126
(V - 62) · (t + 3, 1) = 126
126
V =
t
126
- 62 · (t + 3, 1) = 126
t
126
V =
t
126-62t
· (t + 3, 1) = 126
t
126
V =
t
(126-62t)(t+3,1)
= 126
t
126
V =
t
126t-62t2+390,6-192,2t
= 126/ · t
t
126
V =
t
126t - 62t2 + 390, 6 - 192, 2t = 126t
126
V =
t
-62t2 - 192, 2t + 390, 6 = 0/ : (-62)
126
V =
t
t2 + 3, 1t - 6, 3 = 0
= 3, 12 - 4 · 1 · (-6, 3) = 9, 61 + 25, 2 = 34, 81
"
= 5, 9
-3,1+5,9
t1 = = 1, 4
2
-3,1-5,9
t2 = = -4, 5 t2 < 0 nie spelnia zalożenia
2
t = 1, 4
126
V =
t
t = 1, 4
126
V =
1,4
t = 1, 4
V = 90
km
V = 90
h
km km km km
Vr = V - 62 = 90 - 62 = 28
h h h h
km km
90 28
h h
30cm
20cm
x, y
0 < x < 20 0 < y < 30
ABC <" GF C
20 20-x
=
30 y
20y = 30 (20 - x)
20y = 600 - 30x/ : 20
y = 30 - 1, 5x
P = xy
P (x) = x (30 - 1, 5x) x " (0, 20)
P (x) = -1, 5x2 + 30x
-b
xW =
2a
-30
xW = = 10
2·(-1,5)
xW " (0, 20) P (x) xW
P (x) x = 10.
x = 10
y = 30 - 1, 5 · 10 = 15
10cm × 15cm.
W (x) = x3 + ax2 - 4x + b W (1) = -24 W (-3) = 20
a b
W (x)
W (x) = x3 + ax2 - 4x + b
W (1) = -24
W (1) = 13 + a · 12 - 4 · 1 + b = 1 + a - 4 + b = a + b - 3
a + b - 3 = -24
a + b = -21
W (-3) = 20
W (-3) = (-3)3 + a · (-3)2 - 4 · (-3) + b = -27 + 9a + 12 + b = 9a + b - 15
9a + b - 15 = 20
9a + b = 35
a + b = -21/ · (-1)
9a + b = 35
-a - b = 21
9a + b = 35
8a = 56/ : 8
-a - b = 21
a = 7
b = -21 - a
a = 7
b = -21 - 7
a = 7
b = -28
czyli : W (x) = x3 + 7x2 - 4x - 28
W (x) = x2 (x + 7) - 4 (x + 7)
W (x) = (x + 7) x2 - 4
W (x) = (x + 7) (x - 2) (x + 2)
W (x) = 0
x1 + 7 = 0 x2 - 2 = 0 x3 + 2 = 0
x1 = -7 x2 = 2 x3 = -2
W (x) W (x) = (x + 7) (x - 2) (x + 2)
-7, 2, -2.
x3-x2-12x
W (x) =
x5+3x4-16x3-48x2
x5 + 3x4 - 16x3 - 48x2 = 0

x2 x3 + 3x2 - 16x - 48 = 0

x2 x2 (x + 3) - 16 (x + 3) = 0

x2 (x + 3) x2 - 16 = 0

x2 (x + 3) (x - 4) (x + 4) = 0

x = 0 '" x + 3 = 0 '" x - 4 = 0 '" x + 4 = 0

x = 0 '" x = -3 '" x = 4 '" x = -4

DW = R\ {0, -3, -4, 4}
x3-x2-12x
W (x) =
x5+3x4-16x3-48x2
x x2-x-12
( )
W (x) =
x2(x+3)(x-4)(x+4)
x2 - x - 12
= (-1)2 - 4 · 1 · (-12) = 1 + 48 = 49
"
= 7
1-7 -6 1+7 8
x1 = = = -3 x2 = = = 4
2·1 2 2·1 2
x2 - x - 12 = (x + 3) (x - 4)
x(x+3)(x-4)
W (x) =
x2(x+3)(x-4)(x+4)
1
W (x) =
x(x+4)
W (x) DW = R\ {0, -3, -4, 4}
1
W (x) =
x(x+4)
x6-1
f (x) =
(x2-1)(x5+x4+x3+x2+x+1)
g (x) = |f (x) + 2|
x2 - 1 x5 + x4 + x3 + x2 + x + 1 = 0

(x - 1) (x + 1) x4 (x + 1) + x2 (x + 1) + 1 (x + 1) = 0

(x - 1) (x + 1) (x + 1) x4 + x2 + 1 = 0

(x - 1) (x + 1)2 x4 + x2 + 1 = 0

x - 1 = 0 '" x + 1 = 0 '" x4 + x2 + 1 = 0 x " R

x = 1 '" x = -1

Df = R\ {1, -1}
x6-1 x6-1 x6-1 1
f (x) = = = =
(x2-1)(x5+x4+x3+x2+x+1) (x+1)(x-1)(x5+x4+x3+x2+x+1) (x+1)(x6-1) x+1
1
f (x) =
x+1
g (x) = |f (x) + 2| Dg = R\ {1, -1}
1
g (x) = + 2
x+1
1 1
x2 + x3 + = 18.
x2 x3
x > 0
2
1 1
x + = x2 + 2 +
x x2
2
1 1
x2 + = x + - 2
x2 x
3
1 1 1 1 1 1 1 1
x + = x3 + 3 · x2 · + 3 · x · + = x3 + 3x + 3 + = x3 + + 3 x +
x x x2 x3 x x3 x3 x
3
1 1 1
x + - 3 x + = x3 +
x x x3
1
x3 + = 18
x3
3
1 1
x + - 3 x + = 18
x x
1
k x + k > 0
x
k3 - 3k = 18
k3 - 3k - 18 = 0
k3 + 3k2 + 6k - 3k2 - 9k - 18 = 0
k k2 + 3k + 6 - 3 k2 + 3k + 6 = 0
(k - 3) k2 + 3k + 6 = 0
k - 3 = 0 (" k2 + 3k + 6 = 0
k = 3 = 33 - 4 · 1 · 6 = 9 - 24 = -15 < 0 brak rozwiÄ…zaÅ„
1 1
k = 3 k = x + x + = 3
x x
2
1 1
x2 + = x + - 2
x2 x
1
x2 + = 32 - 2 = 9 - 2 = 7
x2
1
x2 +
x2