ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4

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1 ( ) 8 y = ax 2 + bx + c 443

2 ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444

3 ( a, b, c, d a 0) ( ) yes ( ) ( ) n a n x n + a n 1 x n a 1 x + a 0 =

4 a(x a 1 )(x a 2 ) (x a n ) = 0 x a 1 = 0 x a 2 = 0 x a n = 0 2 x = a 1, a 2,, a n 1 ( ) P (x) x α P (α) = 0 x 3 2x 1 = 0 x 3 2x 1 P (x) = x 3 2x 1 P ( 1) = = 0 P (x) x + 1 P (x) = (x + 1)(x 2 x 1) (x + 1)(x 2 x 1) = 0 x + 1 = 0 x 2 x 1 = 0 x = 1, 1 ± αβ = 0 α = 0 β = 0 α, β αβγ = 0 α = 0 β = 0 γ = 0 446

5 ( ) x 2 x 1 = 0 ( ) x 4 + 3x 3 + 2x 2 2x 4 = 0 P (x) = x 4 + 3x 3 + 2x 2 2x 4 P (1) = 0 P (x) = (x 1)(x 3 + 4x 2 + 6x + 4) Q(x) = x 3 +4x 2 +6x+4 Q( 2) = 0 Q(x) = (x+2)(x 2 +2x+2) (x 1)(x + 2)(x 2 + 2x + 2) = 0 x 1 = 0 x + 2 = 0 x 2 + 2x + 2 = 0 x = 1, 2, 1 ± 3i ( ) 205 (1) x 3 2x 2 4x + 3 = 0 (2) x 4 + x 3 3x 2 4x 4 = 0 (3) x 3 (a + 2)x 2 + (2a + 4)x 4a = 0 ( (3) 4a ) ( ) n 447

6 (n ) n ( ) n x 3 x 2 x + 1 = 0 (x + 1)(x 1) 2 = 0 ( ) x = 1, 1, 1 ( ) x 3 + 3x 2 + 3x + 1 = 0 (x + 1) 3 = 0 x = 1, 1, ( ) ( ) n a n x n + a n 1 x n a 1 x + a 0 = 0 a n (x α 1 ) p1 (x α l ) p l = 0 α 1,, α l p i x = α i p i ( ) 206 (1) x 3 + 6x x + 8 = 0 (2) x 4 8x = 0 448

7 x 3 1 = 0 (x 1)(x 2 1 ± 3i +x+1) = 0 x = 1, 2 x 3 1 = 0 x 3 = 1 1 ± 3i 1, 2 ( ) a x 3 = a a ( ) 1 ± 3i 1, ( ) ax 3 + bx 2 + cx + d = 0 α, β, γ α + β + γ = b a αβ + βγ + γα = c a αβγ = d a ax 3 + bx 2 + cx + d = 0 a(x α)(x β)(x γ) = 0 449

8 a(x α)(x β)(x γ) = ax 3 a(α + β + γ)x 2 + a(αβ + βγ + γα)x aαβγ b = a(α + β + γ) c = a(αβ + βγ + γα) d = aαβγ ( ) (1) (2) ( ) x 3 6x x 8 = 0 α, β, γ α + β + γ = 6 αβ + βγ + γα = 12 αβγ = 8 ( ) 75 x 3 + ax 2 + bx + 24 = 0 a, b a b ( ) 450

9 α α = a 6 + 2α + 3α = b 6α = 24 a, b α α = a 6 + 2α + 3α = b 6α = 24 a = 1, b = 14, α = 4 ( ) ( ) 207 x 3 + ax 2 + bx + 2 = 0 2 a, b n 1 451

10 ax 2 + bx + c a(x α)(x β) α, β ax 2 + bx + c = 0 α, β b 2 4ac 3 x 2 x a, b b 2 4ac a, b x 2 5x + 6 (x 2)(x 3) D D = ( 5) = = 1 x 2 x 5 4 ( ) x 2 + x 1 D = ( 1) = 5 ( ) x 2 + x + 1 D D = = 3 ( ) 208 (1) x 2 12x 1 (2) 3x 2 + 2x + 1 (3) 4x 2 16x + 15 ( ax 2 + 2b x + c = 0 D/4 ( ) )

11 ( ) ( ) ( ) 5 ( ) x 2 4 x 2 4 = (x + 2)(x 2) x 2 2 x 2 2 = (x + 2)(x 2) 2 x x = (x + 2i)(x 2i) 2i x 2 4 x 2 2 x x 2 2 x ( ) 5 453

12 20.2 ( ) 8 76 { x y = 4 x 2 3y 2 = 6 { x y = 4 (1) x 2 3y 2 = 6 (2) (1) x = y + 4 (1) (2) y 2 4y 5 = 0 y = 1, 5 ( ) y = 1 (1) x = 3 ( ) y = 5 (1) x = 9 x = 3, y = 1 x = 9, y = 5 ( ) ( ) 209 { { 2x y = 1 x + y = 5 (1) (2) x 2 y 2 = 1 x 2 + xy + y 2 =

13 77 { x + y = 2 xy = 4 x, y t 2 2t + 4 = 0 ( ) x, y t 2 2t + 4 = 0 t = 1 ± 3i x = 1 ± 3i, y = 1 3i ( ) ( ) (1) x y (2) x = 1 ± 3i, y = 1 3i ( ) 455

14 ± ( ) x y x x = 1 + 3i y y = 1 3i x = 1 ± 3i, y = 1 3i ( ) 210 { { x + y = 1 x + y = 2 (1) (2) xy = 42 xy = 2 (3) { x + y = 3 (x 3)(y 3) = 2 78 { x 2 y 2 = 1 2x 2 + 3xy 2y 2 = 0 (x + 2y)(2x y) = 0 x + 2y = 0 2x y = 0 { f(x, y) = 0 g(x, y) = 0 456

15 f(x, y) = 0 g(x, y) = 0 f(α, β) = 0 g(α, β) = 0 (α, β) g 1 (x, y) = 0 g 2 (x, y) = 0 f(α, β) = 0 (g 1 (α, β) = 0 g 2 (α, β) = 0) (α, β) ( ) p (q 1 q 2 ) 6 p (q 1 q 2 ) ( ) (p q 1 ) (p q 2 ) { x 2 y 2 = 1 x + 2y = 0 { x 2 y 2 = 1 2x y = 0 x + 2y = 0 ( ) x + 2y = 0 (x + 2y)(2x y) = 0 2x y = 0 x = 2y x = ± , y = ( ) 3 ( ) 2x y = 0 3 y = 2x x = ± 3 i, y = ± i ( ) ( ) 211 { 5x 2 4xy y 2 = 0 x 2 + y 2 = 2 457

16 79 { x 2 xy = 4x + 2y y 2 xy = 2x + y x 2 + xy 2y 2 = 0 { x 2 xy = 4x + 2y y 2 xy = 2x + y { x 2 xy = 4x + 2y x 2 + xy 2y 2 = 0 { { x 2 xy = 4x + 2y x 2 xy = 4x + 2y 116 y 2 xy = 2x + y x 2 + xy 2y 2 = 0 { x 2 xy = 4x + 2y (1) y 2 xy = 2x + y (2) (1) 2 (2) x 2 + xy 2y 2 = 0 (3) { x 2 xy = 4x + 2y x 2 + xy 2y 2 = 0 (3) x y = 0 x + 2y = 0 458

17 ( ) x y = 0 (1) x = 0, y = 0 ( ) x + 2y = 0 (1) x = 0, y = 0 x = 2, y = 1 ( ) ( ) ( ) 212 { x x 2 2xy = 0 y y 2 2xy = 0 ( 2xy ) x 1 6 x 2 1 = 1 x 1, (x + 1)(x 1) (x + 1)(x 1) 3(x + 1) 6 = x

18 6 3(x + 1) 6 = x 2 1 x = 1, 2 x = 1 x = 2 ( ) ( ) x x 2 9 = 0 x 2 x x 3 = x 3(x + 3) + 3 x x = ±3 ( ) ( ) A = B (A, B x ) 6 460

19 81 1 x = x + 1 ( 1 x) 2 a = b = a 2 = b 2 ( ) 1 x = (x + 1) 2 x = 0, 3 ( ) x = 0 ( ) = 1 0 = 1 ( ) = = 1 ( ) x = 3 ( ) = 1 ( 3) = 2 ( ) = 0 + ( 3) = 2 x = 0 ( ) ( ) 214 (1) x + 2 = x (2) 8 x = 2 x (3) 2x x = 0 461

20 20.4 z 2 = 1 + 3i z (x + yi) 2 = 1 + 3i x, y (x + yi) 2 = x 2 y 2 + 2xyi a + bi c + di a = c, b = d { x 2 y 2 = 1 2xy = 3 x, y 3 2x y = 2x x 2 3 4x 2 = 1 4x 2 4x 4 4x 2 3 = 0 x 2 (2x 2 3)(2x 2 + 1) = 0 x 2 = 3 2, 1 2 x x 2 > = 0 x 2 = 3 2 x = ±

21 y = 3 2x y y = ± 6 2 z = ± 2 ± ( ) i 6 2 i z = ± 2 ± 2 i ( ) ( ) a + bi (b 0) ( ) (x + yi) 2 = a + bi x, y { x 2 y 2 = a 2xy = b 2xy = b b x, y ( ) 2x y = b 2x x 2 b2 4x 2 = a 4x 2 4x 4 4ax 2 b 2 = 0 x x 2 x 2 D D/4 = ( 2a) 2 4 ( b 2 ) = 4a 2 + 4b 2 = 4(a 2 + b 2 ) 463

22 a, b b 0 a 2 + b 2 > 0 D > 0 x 2 4x 4 4ax 2 b 2 = 0 x 2 = 2a ± 4(a 2 + b 2 ) 4 = a ± (a 2 + b 2 ) 2 a < (a 2 + b 2 ) x x 2 > = 0 x 2 = a + (a 2 + b 2 ) 2 a + (a x 2 + b 2 ) 2 y = b 2x y (x + yi) 2 = a + bi x, y ( ) a + bi i 20.5 III

23 1976 G. S

> > <., 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

> > <., 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 13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 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 >

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