Завдання 626 Квадрат суми або квадрат різниці двох виразів
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1) (a + 50)²; Так 2) a² + b²; Ні |
3) (5 – x)²; Так 4) m² – n²; Ні |
5) (xy + mn)²; Так 6) (6 – c)3? Ні |
Завдання 627
(5a + 3)² = (5a)² + 2 • 3 • 5a + 3² = 25a² + 30a + 9, тому 2) 25a² + 30a + 9
Завдання 628 Тотожність
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1) (12a – b)² = 144a² – b²; Ні 2) (12a – b)² = 144a² + 24ab + b²; Ні |
3) (12a – b)² = 144a² – 24ab + b²; Так 4) (12a – b)² = 12a² – 24ab + b²? Ні |
Завдання 629 Многочлен
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1) (a + x)² = a² + 2ax + x² 2) (x + 2)² = x² + 4x + 4 3) (y – 1)² = y² – 2y + 1 |
4) (5 – p)² = 25 – 10p + p² 5) (y – 13)² = y² – 26y + 169 6) (13 – y)² = 169 – 26y + y² |
Завдання 630 Піднесення до квадрата
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1) (a + 8)² = a² + 16a + 64 2) (b – 2)² = b² – 4b + 4 3) (7 + c)² = 49 + 14c + c² |
4) (4 + k)² = 16 + 8k + k² 5) (6 – d)² = 36 – 12d + d² 6) (d – 6)² = d² – 12d + 36 |
Завдання 631
1) (3x + 5y)² = (3x)² + 2 • 3x • 5y + (5y)² = 9x² + 30xy + 25y²
2) (1/2 a + 6b)² = (1/2 a)² + 2 • 1/2 a • 6b + (6b)² = 1/4 a² + 6ab + 36b²
3) (1/3 x4 + 0,6y5)2 = (1/3 x4)2 + 2 • 1/3 x4 • 0,6y5 + (0,6y5)2 = 1/9 x8 + 0,4x4y5 + 0,36y10
Завдання 632
1) (ab – 9)² = (ab)² – 2 • ab • 9 + 9² = a²b² – 18ab + 81
2) (4a2 – a3)2 = (4a2)2 – 2 • 4a² • a3 + (a3)2 = 16a4 – 8a5 + a6
Завдання 633
1) (3a – 2)² = (3a)² – 2 • 3a • 2 + 2² = 9a² – 12a + 4
2) (7b + 6)² = (7b)² + 2 • 7b • 6 + 6² = 49b² + 84b + 36
3) (8x + 4y)² = (8x)² + 2 • 8x • 4y + (4y)² = 64x² + 64xy + 16y²
4) (0,4m – 0,5n)² = (0,4m)² – 2 • 0,4m • 0,5n + (0,5n)² = 0,16m² – 0,4mn + 0,25n²
5) (3a + 1/3 b)² = (3a)² + 2 • 3a • 1/3 b + (1/3 b)² = 9a² + 2ab + 1/9 b²
6) (b2 – 11)2 = (b2)2 – 2 • b2 • 11 + 11² = b4 – 22b2 + 121
7) (a2 + 4b)2 = (a2)2 + 2 • a2 • 4b + (4b)2 = a4 + 8a2b + 16b2
8) (a2 + a)2 = (a2)2 + 2 • a2 • a + a2 = a4 + 2a3 + a2
9) (3b2 – 2b5)2 = (3b2)2 – 2 • 3b2 • 2b5 + (2b5)2 = 9b4 – 12b7 + 4b10
Завдання 634
1) (2m + 1)² = (2m)² + 2 • 2m • 1 + 1² = 4m² + 4m + 1
2) (4x – 3)² = (4x)² – 2 • 4x • 3 + 3² = 16x² – 24x + 9
3) (10c + 7d)² = (10c)² + 2 • 10c • 7d + (7d)² = 100c² + 140cd + 49d²
4) (4x – 1/8 y)² = (4x)² – 2 • 4x • 1/8 y + (1/8 y)² = 16x² – xy + 1/64 y²
5) (0,3a + 0,9b)² = (0,3a)² + 2 • 0,3a • 0,9b + (0,9b)² = 0,09a² + 0,54ab + 0,81b²
6) (c2 – 6)2 = (c2)2 – 2 • c2 • 6 + 62 = c4 – 12c2 + 36
7) (m2 – 3n)2 = (m2)2 – 2 • m2 • 3n + (3n)2 = m4 – 6m2n + 9n2
8) (m4 – n3)2= (m4)2 – 2 • m4 • n3 + (n3)2 = m8 – 2m4n3 + n6
9) (5a4 – 2a7)2 = (5a4)2 – 2 • 5a4 • 2a7 + (2a7)2 = 25a8 – 2011 + 4a14
Завдання 635 Спростіть вираз
1) a² + (3a – b)² = a² + (3a)² – 2 • 3a • b + b² = a² + 9a² – 6ab + b² = 10a² – 6ab + b²
2) (4x + 5)² – 40x = (4x)² + 2 • 4x • 5 + 5² – 40x = 16x² + 40x + 25 – 40x = 16x² + 25
3) 50a² – (7a – 1)² = 50a² – (7a)² + 2 • 7a • 1 – 1² = 50a² – 49a² + 14a – 1 = a² + 14a – 1
4) c² + 36 – (c – 6)² = c² + 36 – (c² – 12c + 36) = c² + 36 – c² + 12c – 36 = 12c
5) (x – 2)² + x(x + 10) = (x² – 4x + 4) + (x² + 10x) = 2x² + 6x + 4
6) 3m(m – 4) – (m + 2)² = 3m² – 12m – (m² + 4m + 4) = 3m² – 12m – m² – 4m – 4 =
= 2m² – 16m – 4
Завдання 636
1) (x – 12)² + 24x = (x² – 24x + 144) + 24x = x² – 24x + 144 + 24x = x² + 144
2) (x + 8)² – x(x + 5) = (x² + 16x + 64) – (x² + 5x) = x² + 16x + 64 – x² – 5x = 11x + 64
3) 2x(x + 2) – (x – 2)² = (2x² + 4x) – (x² – 4x + 4) = 2x² + 4x – x² + 4x – 4 = x² + 8x – 4
4) p(p – 7) – (p + 7)² = (p² – 7p) – (p² + 14p + 49) = p² – 7p – p² – 14p – 49 = –21p – 49
Завдання 637
Доведіть тотожність (a – b)² = (b – a)².
(a – b)² = a² – 2ab + b²
(b – a)² = b² – 2ab + a²
a² – 2ab + b² = b² – 2ab + a²
(a – b)² = (b – a)²
Завдання 638
1) (y – 9)² + (4 – y)(y + 6) = (y² – 18y + 81) + (4y + 24 – y² – 6 y) =
= y² – 18y + 81 + 4y + 24 – y² – 6y = –20y + 105
2) (x – 4)(x + 4) – (x – 1)² = (x² – 16) – (x² – 2x + 1) = x² – 16 – x² + 2x – 1 = 2x – 17
3) (2a – 3b)² + (3a + 2b)² = (4a² – 12ab + 9b²) + (9a² + 12ab + 4b²) =
= 4a² – 12ab + 9b² + 9a² + 12ab + 4b² = 13a² + 13b²
4) (x – 5)² – (x – 7)(x + 7) = (x² – 10x + 25) – (x² – 49) = x² – 10x + 25 – x² + 49 = –10x + 74
Завдання 639
1) (y + 7)² + (y + 2)(y – 7) = (y² + 14y + 49) + (y² – 5y – 14) = y² + 14y + 49 + y² – 5y – 14=
= 2y² + 9y + 35
2) (a + 1)(a – 1) – (a + 4)² = (a² – 1) – (a² + 8a + 16) = a² – 1 – a² – 8a – 16 = –8a – 17
3) (x – 10)(9 – x) + (x + 10)² = (x – 10)(–x + 9) + (x² + 20x + 100) =
= –x² + 19x – 90 + x² + 20x + 100 = 39x + 10
4) (x – 4)(3 + x) – (x – 3)² = (x² – x – 12) – (x² – 6x + 9) = x² – x – 12 – x² + 6x – 9 = 5x – 21
Завдання 640 Рівняння
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1) (x – 8)² – x(x + 6) = –2 x² – 16x + 64 – x² – 6x = –2 –22x = –2 – 64 –22x = –66 x = 3 |
3) (2x + 1)² – (2x – 1)(2x + 3) = 0 4x² + 4x + 1 – (4x² + 6x – 2x – 3) = 0 4x² + 4x + 1 – 4x² – 6x + 2x + 3 = 0 4x² + 4x – 4x² – 6x + 2x = –1 – 3 0x = –4 Немає коренів |
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2) (x + 7)² = (x – 3)(x + 3) x² + 14x + 49 = x² – 9 x² + 14x – x² = –9 – 49 14x = –58 x = –29/7 = –3 1/7 |
4) x(x – 2) – (x + 5)² = 35 x² – 2x – (x² + 10x + 25) = 35 x² – 2x – x² – 10x – 25 = 35 x² – 2x – x² – 10x = 35 + 25 –12x = 60 x = –5 |
Завдання 641
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1) (x + 9)² – x(x + 8) = 1 x² + 18x + 81 – x² – 8x = 1 x² + 18x – x² – 8x = 1 –81 10x = –80 x = –8 |
3) (x – 4) (x + 4) – (x + 6)² = –16 x² – 16 – (x² + 12x + 36) = –16 x² – 16 – x² – 12x – 36 = –16 x² – x² – 12x = –16 + 16 + 36 –12x = 36 x = –3 |
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2) (x – 11)² = (x – 7) (x – 9) x² – 22x + 121 = x² – 9x – 7x + 63 x² – 22x – x² + 9x + 7x = 63 – 121 –6x = –58 x = 29/3 = 9 2/3 |
4) (1 – 3x)² – x(9x – 2) = 5 1 – 6x + 9x² – (9x² – 2x) = 5 1 – 6x + 9x² – 9x² + 2x = 5 –6x + 9x² – 9x² + 2x = 5 – 1 –4x = 4 x = –1 |
Завдання 642
Замініть зірочки такими одночленами, щоб утворилася тотожність:
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1) (2a + b)² = 4a² + 4ab + b² 2) (4x – 10y)² = 16x² – 80xy + 100y² |
3) (2b2 – 5c)2 = 4b4 – 20b2c + 25c2 4) (7a2 + 3b3)2 = 49a4 + 42a2b3 + 9b6 |
Завдання 642
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1) (2a + 6b)² = 4a² + 24ab + 36b² |
2) (3m2 – 7n8)2 = 9m4 – 42m2n8 + 49n16 |
Завдання 644
1) (–x + 1)² = (–x)² + 2(–x)(1) + (1)² = x² – 2x + 1
2) (–m – 9)² = (–m)² + 2(–m)(–9) + (–9)² = m² + 18m + 81
3) (–5a + 3b)² = (–5a)² + 2(–5a)(3b) + (3b)² = 25a² – 30ab + 9b²
4) (–4x – 8y)² = (–4x)² + 2(–4x)(–8y) + (–8y)² = 16x² + 64xy + 64y²
5) (–0,7c – 10d)² = (–0,7c)² + 2(–0,7c)(–10d) + (–10d)² = 0,49c² + 14cd + 100d²
6) (–4a2 + 1/8 ab)2 = (–4a2)2 + 2(–4a²)(1/8 ab) + (1/8 ab)2 = 16a4 – a3b + 1/64a2b2
Завдання 645
1) (–3m + 7n)² = (–3m)² + 2(–3m)(7n) + (7n)² = 9m² – 42mn + 49n²
2) (–0,4x – 1,5y)² = (–0,4x)² + 2(–0,4x)(–1,5y) + (–1,5y)² = 0,16x² + 1,2xy + 2,25y²
3) (–x2 – y)2 = (–x2)2 + 2(–x2)(–y) + (–y)2 = x4 + 2x2y + y2
4) (–a2b2 + c10)2 = (–a2b2)2 + 2(–a2b2)(c10) + (c10)2 = a4b4 – 2a2b2c10 + c20
Завдання 646
1) (10a2 – 7ab2)2 = (10a2)2 – 2(10a2)(7ab2) + (7ab2)2 = 100a4 – 140a3b2 + 49a2b4
2) (0,8b3 + 0,2b2c4)2 = (0,8b3)2 + 2(0,8b3)(0,2b2c4) + (0,2b2c4)2 =
= 0,64b6 + 0,32b5c4 + 0,04b4c8
3) (1 1/3 a2b + 2 1/4 ab2)2 = (4/3 a2b + 9/4 ab2)2 = 16/9 a4b2 + 6a3b3 + 81/16 a2b4 =
= 1 7/9 a4b2 + 6a3b3 + 5 1/16 a2b4
4) (2 1/3 x3y2 – 9/14 y8x)2 = (7/3 x3y2 – 9/14 y8x)2 = 49/9 x6y4 – 3x4y10 + 81/196 y16x2 =
= 5 4/9 x6y4 – 3x4y10 + 81/196 y16x2
Завдання 647
1) 6(1 – 2c)² = 6(1 – 4c + 4c²) = 6 – 24c + 24c²
2) –12(x + 1/3y)² = –12(x² + 2/3xy + 1/9y²) = –12x² – 8xy – 4/3y²
3) a(a – 6b)2 = a(a2 – 12ab + 36b2) = a3 – 12a2b + 36ab2
4) 5b(b2 + 7b)2 = 5b(b4 + 14b3 + 49b2) = 5b5 + 70b4 + 245b3
5) (a + 3)(a – 4)2 = (a + 3)(a2 – 8a + 16) = a3 – 8a2 + 16a + 3a2 – 24a + 48 =
= a3 – 5a2 – 8a + 48
6) (2x + 4)2(x – 8) = (4x2 + 16x + 16)(x – 8) = 4x3 – 32x2 + 16x2 – 128x + 16x – 128 =
= 4x3 – 16x2 – 112x + 128
7) (a – 5)2(a + 5)2 = ((a – 5)(a + 5))2 = (a2 – 25)2 = a4 – 50a2 + 625
8) (3x + 4y)2(3x – 4y)2 = ((3x)2 – (4y)2)2 = (9x2 – 16y2)2 = 81x4 – 288x2y2 + 256y4
Завдання 648
1) (0,02p3k + 20p2k4)2 = (0,02p3k)2 + 2(0,02p3k)(20p2k4) + (20p2k4)2 =
= 0,0004p6k2 + 0,8p5k5 + 400p4k8
2) (1 1/6 mn – 4/21 m2m5)2 = (7/6 mn – 4/21 m2n5)2 = 49/36 m2n2 – 4/9 m3n6 + 16/441 m4n10 =
= 1 13/36 m2n2 – 4/9 m3n6 + 16/441 m4n10
3) –15(1/3 a – 1/5 b)2 = –15(1/9 a2 – 2/15 ab + 1/5 b2) = –5/3 a2 + 2ab – 3/5 b2 =
= –1 2/3 a2 + 2ab – 3/5 b2
4) 7x(x3 – 2x)2 = 7x(x6 – 4x4 + 4x2) = 7x7 – 28x5 + 28x3
5) (5y – 2)2(2y + 1) = (25y2 – 20y + 4)(2y + 1) = 50y3 + 25y2 – 40y2 – 20y + 8y + 4 =
= 50y3 – 15y2 – 12y + 4
6) (10p – k)2(10p + k)2 = ((10p)2 – k2)2 = (100p2 – k2)2 = 10 000p4 – 200p2k2 + k4
Завдання 649 Вирази
1) (a + 3)² –(a – 9)(a + 9) = (a² + 6a + 9) – (a² – 81) = a² + 6a + 9 – a² + 81 = 6a + 90
Якщо a = –2,5, тоді 6(–2,5) + 90 = –15 + 90 = 75
2) (5x – 8)² –(4x – 3)² + 26x = (25x² – 80x + 64) – (16x² – 24x + 9) + 26x =
= 25x² – 80x + 64 – 16x² + 24x – 9 + 26x = 9x² – 30x + 55
Якщо x = –1/3, тоді 9(–1/3)² – 30(–1/3) + 55 = 9(1/9) + 10 + 55 = 1 + 10 + 55 = 66
3) (3y2 + 4)2 + (3y2 – 4)2 – 2(1 – 3y2)(1 + 3y2) =
= (9y4 + 24y2 + 16) + (9y4 – 24y2 + 16) – 2(1 – 9y4) = 18y4 + 32 – 2 + 18y4 = 36y4 + 30
Якщо y = 1/2, тоді 36(1/2)4 + 30 = 36 • 1/16 + 30 = 36/16 + 30 = 9/4 + 30 = 129/4 = 32,25
Завдання 650
1) 2m(m – 6)2 – m2(2m – 15) = 2m(m2 – 12m + 36) – m2(2m – 15) =
= 2m3 – 24m2 + 72m – (2m3 – 15m2) = 2m3 – 24m2 + 72m – 2m3 + 15m2 = –9m2 + 72m
Якщо m = –4, тоді –9 • (–4)² + 72 • (–4) = –9 • (16) – 288 = –144 – 288 = –432
2) (2x – 5)² – 4(x + 1)(x – 7) = (4x² – 20x + 25) – 4(x² – 6x – 7) =
= 4x² – 20x + 25 – 4x² + 24x + 28 = 4x + 53
Якщо x = –3,5, тоді 4 • (–3,5) + 53 = –14 + 53 = 39