> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3
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1 ( ) ( ) ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275
2 > > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3) D < 0 x ( a ) 276
3 D > 0 D= 0 D < y x x x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3) D < 0 x 277
4 x 2 5x + 6 > 0 x y = x 2 5x + 6 x y x ( ) y = x 2 6x + 5 x 2 x 2 5x + 6 = 0 x = 2, 3 y = x 2 5x + 6 x x y = x 2 5x + 6 x y O 2 3 x x 2 5x + 6 > 0 y y x y x x < 2 3 < x ( x ) x = 2 x = 3 y x < 2 3 < x x 2 2x + 3 > = 0 278
5 y = x 2 2x + 3 x x 2 2x + 3 = 0 x = 3, 1 >= 0 x x x 2 2x + 3 > = 0 3 < = x < = 1 x y > 0 y > = 0 y < 0 y < = 0 x x ( ) x x ( ) 2 ax 2 + bx + c > 0 279
6 3 1 x (1) 2 ax 2 + bx + c = 0 (2) y = ax 2 + bx + c x (3) 122 (1) x 2 2x 8 < 0 (2) x < = 0 (3) x 2 + 2x 4 < = 0 ( ) x x x 2 4x + 4 > 0 y = x 2 4x+4 x 2 4x+4 = 0 x = 2 x y > 0 ( ) x 2 4x + 4 > 0 x = 2 x 2 x < 2 2 < x ( ) 280
7 2 2 x 2 4x + 4 > = 0 x x x = 2 x 2 4x + 4 > = 0 x 2 4x + 4 < = 0 x ( x ) x = 2 ( ) x 2 4x + 4 < = 0 x = 2 x < a ( ) 281
8 x x 2 4x + 4 < 0 x x 2 4x + 4 < 0 x x 2 + x + 1 > 0 x 2 + x + 1 = 0 x = 1 ± 3 2 D y = x 2 + x + 1 x y > 0 x ( ) x 2 + x + 1 > 0 x 2 + x + 1 < 0 y < 0 x x 2 + x + 1 < x 2 + x + 1 > = 0 x 2 + x + 1 < = 0 282
9 2 x 2 x >, <,, 24 x (1) 9x 2 12x + 4 > 0 (2) 2x 2 3x + 6 < = 0 (3) 2x 2 3x + 2 > 0 (4) 2x 2 5x + 3 > = 0 (5) x 2 + 4x 1 < 0 (6) 9x 2 6x + 1 < = { x 2 + x 72 > 0 2x + 3 > = 0 2 < x < 3 x > = 3 2 x 283
10 x 3 2 < = x < 3 2 < x < 3 x > = x 3 2 < = x < 3 ( ) ( ) 40 { x 2 x 6 < 0 x 2 x 2 > = 0 (1) (2) (3) 284
11 2 < x < 3 x < = 1 2 < = x x 2 < x < = 1 2 < = x < 3 ( ) ( ) 125 { x 2 2x 3 < 0 x 2 + x 2 > = x x 2 + (k + 1)x + 2k 1 = 0 k 285
12 x 2 + (k + 1)x + 2k 1 = 0 D D = (k + 1) 2 4(2k 1) D = k 2 6k + 5 D < 0 k 2 6k + 5 < 0 1 < k < 5 ( ) ( ) 126 x 2 2(k + 3)x 4k = 0 k ( D D > 0 ) x 2 + 2mx + m + 2 = 0 m x 2 + ax + 3a = 0, x 2 ax + a 2 1 = 0 a D 1 D 1 0 a 2 12a 0 a 2 4(a 2 1) 0 286
13 2 3 3 a 0 ( ) ( ) 128 x 2 + 2ax 2a = 0, x 2 + (a 1)x + a 2 = 0 a 13.6 ( ) αβ = 0 α = 0 β = 0 ( ) a, b ab > 0 a > 0 b > 0 a < 0 b < 0 ab < 0 a > 0 b < 0 a > 0 b < 0 ab > 0 a > 0 b > 0 a < 0 b < 0 = ab > 0 a a > 0, a = 0, a < 0 a = 0 ab = 0 a > 0 a < 0 ( ) a > 0 1 a > 0 1 ab > 0 1 a 1 a ab > 1 a
14 b > 0 a > 0 b > 0 ( ) a < 0 1 a < a > 0 b < 0 = ( ) a > 0, b > 0 ( ) ab > 0 ( ) a < 0, b < 0 a > 0, b > 0 ( ) ab > 0 66 ab < 0 a > 0 b < 0 a > 0 b < 0 ( ) > =, < = ( ) 67 abc > 0 ( a > 0 a < 0 ) 1 a > 0, 1 a = 0, 1 a < 0 1 a = 0 a 1 = 0 1 a < 0 a 1 < 0 1 a > 0 288
15 x 2 5x + 6 > 0 x 2 5x + 6 (x 2)(x 3) (x 2)(x 3) > 0 x 2 > 0 x 3 > 0 x 2 < 0 x 3 < 0 ( ) x 2 > 0 x 3 > 0 x > 2 x > 3 x > 3 ( ) x 2 < 0 x 3 < 0 x < 2 x < 3 x < 2 x 2 > 0 x 3 > 0 x 2 < 0 x 3 < 0 x < 2 3 < x x 2 2x + 3 > = 0 1 x 2 + 2x 3 < = 0 (x 1)(x + 3) < = 0 x 1 > = 0 x + 3 < = 0 x 1 < = 0 x + 3 > = 0 ( ) x 1 > = 0 x + 3 < = 0 x > = 1 x < = 3 ( ) x ( ) ( ) x 1 < = 0 x + 3 > = 0 x < = 1 x > = 3 3 < = x < = 1 ( α, β ) 289
16 ( ) ax 2 + bx + c ( a > 0) a(x α)(x β) ( α, β α < β ) ax 2 + bx + c > 0 x < α β < x ax 2 + bx + c < 0 α < x < β ax 2 + bx + c > 0 a(x α)(x β) a(x α)(x β) > 0 a > 0 1 a ( ) (x α)(x β) > 0 x α > 0 x β > 0 x α < 0 x β < 0 x > α x > β x < α x < β α < β x > α x > β x > β x < α x < β x < α ax 2 + bx + c > 0 x < α x > β 68 ax 2 + bx + c < 0 α < x < β ( ) (1) a > 0 x 2 1 a (2) ax 2 + bx + c (3) > = ( ) ( ) 290
17 x 2 x + 2 < 0 1 x 2 + x 2 > 0 (x + 2)(x 1) (x + 2)(x 1) > 0 x < 2 1 < x 129 (1) x 2 6x 8 < 0 (2) x < = 0 x 2 4x + 4 > 0 x 2 4x + 4 = (x 2) 2 α, β (x 2) 2 > 0 a a 2 > = 0 a a 2 > = 0 a 2 = 0 a = 0 a a 2 > = 0 a 2 > 0 a 0 (x 2) 2 > 0 x 2 0 x 2 x 2 4x + 4 > = 0 (x 2) 2 > = 0 291
18 x 2 4x + 4 < = 0 (x 2) 2 < = 0 x (x 2) 2 < 0 (x 2) 2 = 0 x = 2 x 2 4x + 4 < 0 (x 2) 2 < 0 69 ( ) ax 2 + bx + c ( a > 0 ) a(x α) 2 α ax 2 + bx + c > 0 x α ax 2 + bx + c < 0 70 >=, < = x 2 + x + 1 > 0 x 2 + x + 1 = 0 x 2 + x + 1 x 2 + x + 1 x 2 + x + 1 = x 2 + x ( = x + 1 ) ( x + 1 ) > 0 ( x + 1 ) > 0 x x 2 + x + 1 > 0 292
19 x 2 + x + 1 < 0 ( x + 1 ) < 0 x 71 x 2 + x + 1 > = 0 x 2 + x + 1 < = 0 72 (1) ( ) ax 2 + bx + c ( a > 0 ) ax 2 + bx + c > 0 ax 2 + bx + c > 0 (2) >=, < = 130 (1) 9x 2 12x + 4 > 0 (2) 2x 2 3x + 6 < = 0 (3) 2x 2 3x + 2 > 0 (4) 2x 2 5x + 3 > = 0 (5) x 2 + 4x 1 < 0 (6) 9x 2 6x + 1 < = ( ) 293
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漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
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