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(a + b)2 = a2 + b2 + 2ab.BECbaKLAFMDbaSAKLF = (a – b)2, SABCD = a2; SECML = b2; SKBCM = SECDF = ab.Тогда SAKLF = SABCD – 2SKBCM + SECML, т.е. (a + b)2 = a2 – 2ab + b2.106№862.а) (2x + 3)2 = 4x2 + 12x + 9;в) (10 + 8k)2 = 100 + 160k + 64k2;2б) (7y – 6)2 = 49y2 – 84y + 36;г) (5y – 4x)2 = 25y2 – 40xy + 16x2;21 21 ⎞1 2⎛⎛1⎞д) ⎜ 5a + b ⎟ = 25a2 + 2ab +b ; е) ⎜ m − 2n ⎟ =m – mn + 4n2;5 ⎠25⎝⎝4⎠ 16ж) (0,3x–0,5a)2=0,09x2–0,3ax+0,25a2; з) (10c+0,1y)2= 100c2 +2cy + 0,01y2.№863.а) (7 – 8b)2 = 49 – 112b + 64 b2;б) (0,6 + 2x)2 = 0,36 + 2,4x + 4x2;221 211 ⎞⎛1⎞⎛в) ⎜ x − 3 y ⎟ = x2 – 2xy + 9y2;г) ⎜ 4a + b ⎟ =16a2 + ab +b ;98 ⎠64⎝⎝3⎠е) (12a – 0,3c)2 = 144a2 – 7,2ac + 0,09c2.д) (0,1m+5n)2=0,01m2+mn+25n2;№864.а) (–x + 5)2 = x2 – 10x + 25;б) (–z – 2)2 = z2 + 4z + 4;22в) (–n + 4) = n – 8n + 16;г) (–m – 10)2 = m2 + 20mn + 100.№865.а) (x+y)2 = (y + x)2 = (–x – y)2;б) (x – y)2 = (–y + x)2 = (–x + y)2 = (у – х)2.№866.
а) (a – b)2 = a2 – 2ab + b2 = b2 – 2ba + a2 = (b – a)2;б) (–a – b)2 = (–a)2 + 2(–a)(–b) + (–b)2 = a2 + 2ab + b2 = (a + b)2.№867.а) (–9a+4b)2 = 81a2 – 72ab + 16b2;б) (–11x – 7y)2 = 121x2 + 154xy + 49 y2;2⎛ 1⎞ 16в) (–0,8x–05b)2=0,64x2+0,8xb+0,25b2; г) ⎜ −1 p + 6q ⎟ = p2–16pq+36q2;9⎝ 3⎠д) (0,08a – 50b)2 = 0,0064a2 – 8ab + 2500b2;е) (–0,5x – 60y)2 = 0,25x2 + 60xy + 3600y2.№868. а) (–3a + 10b)2 = 9a2 – 60ab +100b2; б) (–6m–n)2=36m2+12mn + n2;в) (0,8х – 0,3у)2 = 0,64х – 0,48ху + 0,09у2;21 ⎞21 2⎛b ;г) ⎜ 5a + b ⎟ = 25a 2 + ab +15 ⎠3255⎝д) (–0,2p – 10q)2 = 0,04p2 + 4pq + 100q2;е) (0,8x – 0,1y)2 = 0,64x2 – 0,16xy + 0,01y2.№869.а) (100 + 1)2 = 10000 + 200 + 1 = 10201; б) (100–1)2 = 10000 – 200 + 1 = 9801;в) 612 = (60 + 1)2 = 3600 + 120 + 1 = 3721;г) 1992 = (200 – 1)2 = 40000 – 400 + 1 = 39601;д) 9992 = (1000 – 1)2 = 1000000 – 2000 + 1 = 998001;е) 7022 = (700 + 2)2 = 490000 + 2800 + 4 = 492804;ж) 9,92 = (10 – 0,1)2 = 100 – 2 + 0,01 = 98,01;з)10,2 = (10 + 0,2)2 = 100 + 4 + 0,04 = 104,04.№870.Абсолютная погрешность: |1 + 2α + α2 – (1 + 2α)| = α2;а) (1 + 0,01)2 ≈ 1 + 2 ⋅ 0,01 = 1,03, |1,0201 – 1,02| = 0,0001,0,0001≈ 0,00009 = 0,009%;1,02107б) (1 – 0,02)2 ≈ 1 – 2 ⋅ 0,02 = 0,96, |0,9604 – 0,96| = 0,0004,21 ⎞⎛⎜ −3 x − y ⎟ ≈ 0,0004 = 0,04%;3 ⎠⎝в) 1,052 = (1 + 0,05)2 ≈ 1 + 2 ⋅ 0,05 = 1,1,0,0025≈ 0,0023 = 0,23%;|1,1025 – 1,1| = 0,0025,1,1г) 1,0052 = (1 + 0,005)2 ≈ 1 + 0,01 = 1,01,0,000025≈ 0,000025 = 0,0025%;|1,010025 – 1,01| = 0,000025,1,01д) 0,972 = (1 – 0,03)2 ≈ 1 – 0,06 = 0,94,0,0009≈ 0,00096 = 0,096%;|0,9409 – 0,94| = 0,0009,0,94е) 0,9992 = (1 – 0,001)2 ≈ 1 – 0,002 = 0,998,0,000001≈ 0,000001 = 0,0001%.|0,998001 – 0,998| = 0,000001,0,998№871.а) (x2– 5)2 = x4 – 10x2 + 25;б) (7 – y3)2 = 49 – 14y3 + y6;4 2248г) (–3p + q3) = 9p2 – 6pq3 + q6;в) (2a + b ) = 4a + 4ab + b ;2⎛1⎞ 1д) (5y3–2x2)2 = 25y6 – 20y3x2 + 4x4; е) ⎜ m4 + 9n2 ⎟ = m8 + 6m4n2 + 81n4.⎝3⎠ 9№872.21⎛1⎞б) ⎜ x3 + 6 x ⎟ = x6 + 6x4 + 36x2;4⎝2⎠в) (c2 – 0,7c3)2 = c4 – 1,4c5 + 0,49c6; г) (4y3 – 0,5y2)2 = 16y6 – 4y5 + 0,25y4;а) (a2 – 3a)2 = a4 – 6a3 + 9a2;29⎛ 1⎞д) ⎜1 a5 + 8a 2 ⎟ = a10+24a7+64a4; е) (0,6b–60b2)2=0,36b2–72b3+3600b4;4⎝ 2⎠221 ⎞11 4⎛⎛⎞ж) ⎜ 3ab − a 2 ⎟ = 9a2b2–a3b+a ; з) ⎜12c 4 + a 6c ⎟ =144c8 + 6a6c56 ⎠436⎝⎝⎠1 12 2+ a c;16и)(0,2xy + 0,5x2y2)2 = 0,04x2y2 + 0,2x3y3 + 0,25x4y4.№873.а) (a2 –2b)2 = a4 – 4a2b + 4b2;б) (x3 + 3y4)2 = x6 + 6x3y4 + 9y8;621272г) (15х – x3)2 = 225x2 – 30x4 + x6;в)(7a +12a) =49a +168a +144a ;д) (3y + 8y5)2 = 9y2 + 48y6 + 64 y10;е) (4a3 – 11a2)2 = 16a6 – 88a5 + 121a4.№874.а) (a + 2b)2 = a2 + 4ab + 4b2;б) (3x + a)2 = 9x2 + 6ax + a2;г) (6a2 – 9c)2 = 36a4 – 108a2c + 81c2;в) (10 – 2m)2 = 100 – 40m + 4m2;д) (15y+0,4x3)2=225y2+12x3y+0,16x6; е) (3a + 2,5b)2 = 9a2 + 6,25b2 + 15ab.108№875.а) (12a – 1)2 – 1 = 144a2 – 24a + 1 – 1 = 144a2 – 24a;б) (2a + 6b)2 – 24ab = 4a2 + 24ab + 36b2 – 24ab = 4a2 + 36b2;в) 121 – (11 – 9x)2 = 121 – (121 – 198x + 81x2) = 198 – 81x2;г) a2b2 – (ab – 7)2 = a2b2 – (a2b2 – 14ab + 49) = 14ab – 49;д) b2 + 49 – (b – 7)2 = b2 + 49 – (b2 – 14b + 49) = 14b;е) a4 – 81 – (a2 + 9)2 = a4 – 81 – (a4 +18a2 + 81) = –18a2 – 162.№876.а) 18a + (a – 9)2 = 18a + a2 – 18a + 81 = a2 + 81;б) (5x – 1)2 – 25x2 = 25x2 – 10x + 1 – 25x2 = –10x + 1;в) 4x2 – (2x – 3)2 = 4x2 – (4x2 – 12x + 9) = 12x – 9;г) (a + 2b)2 – 4b2 = a2 + 4ab + 4b2 – 4b2 = a2 + 4ab.№877.а) (x – 3)2 + x(x + 9) = x2 – 6x + 9 + x2 + 9x = 2x2 + 3x + 9;б) (2a + 5)2 – 5(4a + 5) = 4a2 + 20a + 25 – 20a – 25 = 4a2;в) 9b(b – 1) – (3b + 2)2 = 9b2 – 9b – 9b2 – 12b – 4 = –21b – 4;г) (b – 4)2 + (b – 1)(2 – b) = b2 – 8b + 16 – b2 + 3b – 2 = – 5b + 14;д) (a + 3)(5 – a) – (a – 1)2 = –a2 + 2a + 15 – a2 + 2a – 1 = –2a2 + 4a + 14;е) (5 + 2y)(y – 3) – (5 – 2y)2 = 2y2 – y – 15 – 25 + 20y – 4y2=–2y2 + 19y – 40.№878.а) (x – 10)2 – x(x + 80) = x2 – 20x + 100 – x2 – 80x =100 – 100x = 100(1 – x),при x = 0,97, 100(1 – x) = 100(1 – 0,97) = 3;б) (2x + 9)2 – x(4x + 31) = 4x2 + 36x + 81 – 4x2 – 31x = 5x + 81,при x = –16,2, 5x + 81 = 5 ⋅ (–16,2) – 81 = 0;в) (2x + 0,5)2 – (2x – 0,5)2 = 4x2 + 2x + 0,25 – 4x2 + 2x – 0,25 = 4x,при x = –3,5, 4x = 4 ⋅ (–3,5) = –14;г) (0,1x – 8)2 + (0,1x + 8)2 = 0,01x2 – 1,6x+64+0,01x2+1,6x+64=0,02x2 + 128,при x = –10, 0,02x2 + 128 = 0,02 ⋅ (–10)2 + 128 = 130.№879.а) (x – 6)2 – x(x + 8) = 2, x2 – 12x + 36 – x2 – 8x = 2, 20x = 34, x = 1,7;1б) 9x(x + 6) – (3x + 1)2 = 1, 9x2 + 54x – 9x2 – 6x – 1 = 1, 48x = 2, x =;24в) y(y – 1) –(y – 5)2 = 2, y2 – y – y2 + 10y – 25 = 2, 9y = 27, y = 3;г) 16y(2–y)+(4y–5)2=0, 32y–16y2+16y2 – 40y + 25 = 0, –8y=–25, y =3,25.№880.а) (x – 5)2 – x2 = 3, x2 – 10x + 25 – x2 = 3, 10x = 22, x = 2,2;б) (2y + 1)2 – 4y2 = 5, 4y2 + 4y + 1 – 4y2 = 5, 4y = 4, y = 1;в) 9x2 – 1 – (3x – 2)2 = 0, 9x2 – 1 – 9x2 + 12x – 4 = 0, 12x = 5, x =5;12г) x+(5x+2)2=25(1+x2), x + 25x2 + 20x + 4 – 25 – 25x2=0, 21x = 21, x=1.№881.
а) 7(4a – 1)2 = 7(16a2 – 8a + 1) = 112a2 – 56a + 7;б) –3(5y – x)2 = –3(25y2 – 10xy + x2) = –75y2 + 30xy –3x2;⎛1⎝2⎞⎠2⎛1⎝4⎞⎠52в) –10 ⋅ ⎜ b + 2 ⎟ = −10 ⋅ ⎜ b 2 + 2b + 4 ⎟ = − b 2 − 20b − 40 ;109г) 3(a – 1)2 + 8a = 3(a2 – 2a + 1) + 8a = 3a2 – 6a + 3 + 8a = 3a2 + 2a + 3;д) 9c2–4+6(c–2)2=9c2–4 + 6(c2–4c+4)=9c2 – 4 + 6c2 – 24c + 24=15c2– 24c + 20;е) 10ab – 4(2a – b)2 + 6b2 = 10ab – 4(4a2 – 4ab + b2) + 6b2 == 10ab – 16a2 + 16ab – 4b2 + 6b2 = 26ab – 16a2 + 2b2.№882.а) 5(3a + 7)2 = 5(9a2 + 42a + 49) = 45a2 + 210a + 245;б) –6(4 – b)2 = –6(16 – 8b + b2) = –96 + 48b – 6b2;в) –3(2–x)2–10x=–3(4–4x+x2) – 10x = –12 + 12x – 3x2 – 10x=–3x2 + 2x – 12;г) 12a2 – 4(1 – 2a)2 + 8 = 12a2 – 4(1 – 4a + 4a2) + 8 == 12a2 – 4 + 16a – 16a2 + 8 = –4a2 + 16a + 4.№883.а) a(a + 9b)2 = a(a2 + 18ab + 81b2) = a3 + 18a2b + 81ab2;б)6x ⋅ (x2 + 5x)2 = 6x(x4 + 10x3 + 25x2) = 6x5 + 60x4 + 150x3;в) (a+2)(a–1)2=(a+2)(a2–2a+1) = a3 – 2a2 + a + 2a2 – 4a + 2 = a3 – 3a + 2;г) (x–4)(x+2)2=(x–4)(x2+4x+4)=x3 + 4x2 + 4x – 4x2 – 16x – 16=x3 – 12x – 16.№884.а) (a + b)2 + (a – b)2 = a2 + 2ab + b2 + a2 – 2ab + b2 = 2(a2 + b2);б) (a + b)2 – (a – b)2 = a2 + 2ab + b2 – (a2 – 2ab + b2) = 4ab;в) (a + b)2 – 2ab = a2 + 2ab + b2 – 2ab = a2 + b2;г) (a + b)2 – 2b(a + b) = a2 + 2ab – 2ba – 2b2 = a2 – b2.№885.(a + b)3 = (a + b)2 ⋅ (a + b) = (a2 + 2ab + b2) ⋅ (a + b) == a3 + 2a2b + ab2 + a2b + 2ab2 + b3 = a3 + 3a2b + 3ab2 + b3;а) (2x + y)3 = 8x3 + 12x2y + 6xy2 + y3; б) (a+3b)3 = a3 + 9a2b + 27ab2 + 27b3.№886.(a – b)3 = (a – b)2 ⋅ (a – b) = (a2 – 2ab + b2) ⋅ (a – b) == a3 – 2a2b + ab2 – a2b + 2ab2 – b3 = a3 – 3a2b + 3ab2 – b3.№887.а) (x+1)2–120 = (x – 3)2, x2 + 2x + 1 – 120 = x2 – 6x + 9, 8x = 128, x =16;б) (2x + 10)2 = 4(x – 5)2, 4x2 + 40 + 100 = 4(x2 – 10x + 25), 80x = 0, x= 0.№888.
а) сумма квадратов выражений 3a и 5b;б) квадрат суммы выражений 3a и 5b;в) квадрат разности выражений 3a и 5b;г) разность квадратов выражений 3a и 5b.№889. а) (2m)2 – (7n)2; б) (x – 8y)2; в) 3 ⋅ 6a ⋅ b2; г) (a + b)(a – b).№890.a5 + 2a + a4 + 2 = a4(a + 1) + 2(a + 1) = (a +1)(a4 + 2).№891. а) (2a2 – ab)(a + 4b2) = 2a3 + 8a2b2 – a2b – 4ab3;б) (x + 3y)(x – 3y) = x2 – 3xy + 3xy – 9y2 = x2 – 9y2.№892.vtsI поездx км/ч5ч5x кмвсего: 1020 – 170 кмII поезд(x + 10) км/ч5ч5(x + 10) км5x + 170 + 5(x + 10) = 1020, 5x + 170 + 5x + 50 = 1020, 10x = 800,x = 80 (км/ч) — скорость I поезда, 90 км/ч — скорость II поезда.Ответ: 80 км/ч; 90 км/ч.11032. Разложение на множители с помощью формул квадратасуммы и квадрата разности№893.а) x2 + 2xy + y2 = (x + y)2;в) a2 + 12a + 36 = (a + 6)2;д) 1 – 2z + z2 = (1 – z)2;№894.а) 4x2 + 12x + 9 = (2x + 3)2;б) p2 – 2pq + q2 = (p – q)2;г) 64 + 16b + b2 = (8 + b)2;е) n2 + 4n + 4 = (n + 2)2.б) 25b2 + 10b + 1 = (5b + 1)2;2в) 9x2 – 24xy + 16y2 = (3x – 4y)2; г)1 2⎛1⎞m + 4n2 – 2mn = ⎜ m − 2n ⎟ ;4⎝2⎠2д) 10xy + 0,25x2 + 100y2 = (0,5x + 10y)2;№895.а) 81a2 – 18ab + b2 = (9a – b)2;в) 8ab + b2 + 16a2 = (b + 4a)2;д) b2 + 4a2 – 4ab = (b – 2a)2;№896.а) 16a2 + 56a + 49 = (4a + 7)2;е) 9a2 – ab +1 2 ⎛1 ⎞b = ⎜ 3a − b ⎟ .6 ⎠36⎝б) 1 + y2 – 2y = (1 – y)2;г) 100x2 + y2 + 20xy = (10x + y)2;е) 28xy + 49x2 + 4y2 = (7x + 2y)2.б) 36 – 12x + x2 = (6 – x)2;221 211 ⎞⎛1 ⎞ ⎛b = (5a)2 + 2 ⋅ 5a ⋅ b + ⎜ b ⎟ = ⎜ 5a + b ⎟ ;2 ⎠42⎝2 ⎠ ⎝г) 0,01b2 + 2bc + 100c2 = (0,1b)2+2 ⋅ 0,1b ⋅ 10c + (10c)2 = (0,1b + 10c)2.№897.а) (3b + 2a)2 = 9b2 + 12ab + 4a2;б) (3x + 7y)2 = 9x2 + 42xy + 49y2.№898.а) b2 + 20b + 100 = (b + 10)2;б) b2 + 14b + 49 = (b + 7)2;в) 16x2 + 24xy + 9y2 = (4x + 3y)2;г) 9p2 – 42pq + 49q2 = (3p – 7q)2.№899.а) –1+4a–4a2=–(1 – 4a + 4a2) = –(1 – 2a)2; б) –42a+9a2+49=(3a – 7)2;г) –44ax+121a2 + 4x2 = (11a – 2x)2;в) 24ab – 16a2 – 9b2 = –(4a – 3b)2;222д) 4cd – 25c – 0,16d = –(5c – 0,4d) ; е) –0,49x2–1,4xy–y2=–(0,7x+ y)2.№900.
а) y2 – 2y + 1 = (y – 1)2:при y = 101, (y – 1)2 = (101 – 1)2 = 1002 = 10000,при y = –11, (y – 1)2 = (–11 – 1)2 = 122 = 144,при y = 0,6, (y – 1)2 = (0,6 – 1)2 = 0,42 = 0,16;б) 4x2 – 20x + 25 = (2x – 5)2при x = 12,5, (2x – 5)2 = (2 ⋅ 12,5 – 5)2 = 202 = 400,при x = 0, (2x – 5)2 = (2 ⋅ 0 – 5)2 = 25,при x = –2, (2x – 5)2 = (–2 ⋅ 2 – 5)2 = (–9)2 = 81;в) 25a2 + 49 + 70a = (5a + 7)2при a = 0,4, (5a + 7)2 = (5 ⋅ 0,4 + 7)2 = 92 = 81,при a = –2, (5a + 7)2 = (5 ⋅ (–2) + 7)2 = 9,при a = –1,6, (5a + 7)2 = (5 ⋅ (–1,6) + 7)2 = 1.в) 25a2 + 5ab +111№901.а) x2 + 10 > 0 — верно для любого x, т.к. x2 ≥ 0 и 10 > 0;б) x2 + 20x + 100 > 0 — не верно для любого x, т.к.x2 + 20x + 100 = (x + 10)2, и при x = –10 это равно нулю.№902.а) x2 – 30x + 225 = (x – 15)2 ≥ 0; б) – x2 – 2xy – y2 = –(x + y)2 ≤ 0.№903.а) x2 – 16x + 64 = (x – 8)2 ≥ 0;б) 64 + 8x + x2 = x2 + 8x + 16 + 48 = (x + 4)2 + 48 ≥ 0;в) –x2 – 4x – 4 = –(x2 + 4x + 4) = –(x + 2)2 ≤ 0;г) –x2 + 18x – 81 = –(x2 – 18x + 81) = –(x – 9)2 ≤ 0.№904.2а)1 2⎛1⎞x + 3x + 9 = ⎜ x + 3 ⎟ ;4⎝2⎠б) 25a2 – 30ab + 9b2 = (5a – 3b)2;в) p2 – 2p + 4 — нельзя представить;21 2 21 2 ⎛11 ⎞y = ⎜ x + y⎟ ;x + xy +15255 ⎠9⎝3д) 100b2 + 9c2 – 60bc = (3c – 10b)2;е) 49x2 + 12xy + 64y2 — нельзя представить;ж) 81y2 – 16z2 – 72yz — нельзя представить;21 2⎛1⎞a – ab + 4b2 = ⎜ a − 2b ⎟ .з)16⎝4⎠№905.21 4⎛1⎞а) x4 – 8x2y2 + 16y4 = (x2 – 4y2)2; б)x + 2x2a + 16a2 = ⎜ x 2 + 4a ⎟ ;16⎝4⎠г)2в)1 2⎛1⎞a + 2ab2 + b4 = ⎜ a + b2 ⎟ ;4⎝2⎠г) a2x2 – 2abx + b2 = (ax – b)2;2д) 9y2+c2d2+6cdy = (3y + cd)2; е)9 6 2 4 4 25 2 6 ⎛ 3 3 5 3 ⎞a b –a b +a b = ⎜ a b − ab ⎟ .62536⎝5⎠№906.2а) 4a6 – 4a3b2 + b4 = (2a3 – b2)2; б) b8 – a2b4 +1 4 ⎛ 4 1 2⎞a = ⎜b − a ⎟ ;2 ⎠4⎝в) 0,01x4 + y2 – 0,2x2y = (0,1x2 – y)2; г) 9x8 + 4y2 – 12x4y = (3x4 – 2y)2.№907.а) квадрат разности a и 10b; б) разность квадратов a и 10b;в) произведение суммы a и 10b на их разность.№908.21 ⎞⎛а) ⎜ 3a + b ⎟ ; б) (0,5m2) + (5,3n)2; в) 0,6 ⋅ 9y2; г) (8x+4y)⋅(8x – 4y).3 ⎠⎝112№909.а) (x2+4xy–y2)(2y–x)=2x2–x3+8xy2–4x2y – 2y3 + xy2=–x3–2x2y+9xy2 – 2y3;б) (3 – a)(a3 – 4a2 – 5a) = 3a3 – 12a2 – 15a – a4 + 4a3 + 5a2 == –a4 + 7a3 – 7a2 – 15a.№910.а) 4x4 = (2x2)2; б) 0,25a4 = (0,5a2)2; в) 36m6 = (6m3)2;г) a2b4 = (ab2)2; д) 9a4b2 = (3a2b)2; е) 0,16x6y4 = (0,4x3y2)2.№911.а) m3 – m2 – m – 1 = m2(m + 1) – (m + 1) = (m + 1)(m2 – 1);б) 7a3 – a2b – 28a – 4b = 7a(a2 – 4) + b(a2 – 4) = (a2 – 4)(7a + b).§ 13.
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