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1) x 2 − 4 x + x − 4 = 0 ;x+7 = 04) 3x 2 + 12 x − ( x + 4) = 05 x ⋅ ( x − 2 ) + ( x − 2) = 0 ;3x ⋅ (x + 4) − (x + 4) = 0( x − 2) ⋅ (5x + 1) = 0 ;( x + 4) ⋅ (3x − 1) = 0x − 2 = 0;5x + 1 = 0 ;x + 4 = 0; 3x − 1 = 01x1 = − ;5x2 = 2 ;[( x − 3x ) − ( 2 x232x1 = −4;)]− 6 x : ( x − 2) =x2 =13[]=x ⋅ x ⋅ ( x − 3) − 2 ⋅ ( x − 3)x ⋅ ( x − 3) ⋅ ( x − 2)= x ⋅ ( x − 3) = x 2 − 3xx−2x−2349.

1) x 2 + 3x + 2 = x 2 + 2 x + x + 2 = ( x 2 + 2 x) + (x + 2) == x ⋅ ( x + 2) + ( x + 2) = ( x + 2) ⋅ ( x + 1)2) x 2 − 5 x + 6 = x 2 − 2 x − 3x + 6 = x ⋅ (x − 2) − 3 ⋅ (x − 2) == ( x − 2 ) ⋅ ( x − 3)3) x 2 − 7 x − 8 = x 2 − 8 x + x − 8 = x ⋅ ( x − 8) + ( x − 8) = ( x − 8) ⋅ ( x + 1)4) x 2 + 9 x − 10 = x 2 + 10 x − x − 10 = ( x 2 + 10 x) − (x + 10) == x ⋅ ( x + 10) − ( x + 10) = ( x + 10) ⋅ ( x − 1)350.

1) a 3 + 2a 2 − 3 = a3 + 3a 2 − a 2 − 3 = (3a 2 − 3) + (a 3 − a 2 ) == 3(a 2 − 1) + a 2 (a − 1) = 3(a − 1)(a + 1) + a 2 (a − 1) = (a − 1)(3a + 3 + a 2 )2) x 3 − 7 x + 6 = x 3 − x − 6 x + 6 = x ⋅ ( x − 1) ⋅ ( x + 1) − 6 ⋅ ( x − 1) == (x − 1) ⋅ ( x 2 + x − 6) = (x − 1) ⋅ ( x 2 + 3x − 2 x − 6) =[]= ( x − 1) ⋅ x ⋅ ( x + 3) − 2 ⋅ ( x + 3) = ( x − 1) ⋅ ( x + 3) ⋅ ( x − 2)713) a 4 + 2a3 + 1 = a 4 + a3 + a3 + 1 = a3 ⋅ (a + 1) + (a3 + 1) == a3 ⋅ (a + 1) + (a3 + a 2 − a 2 + 1) = a3 ⋅ (a + 1) + a 2 ⋅ (a + 1) − (a 2 − 1) == a 3 ⋅ (a + 1) + a 2 ⋅ (a + 1) − (a − 1) ⋅ (a + 1) = (a + 1) ⋅ (a 3 + a 2 − a + 1)4) 2a 4 − a 2 − 1 = 2a 4 − 2a 2 + a 2 − 1 = 2a 2 ⋅ (a 2 − 1) + (a 2 + 1) == (a 2 − 1) ⋅ (2a 2 + 1) = (a − 1) ⋅ (2a 2 + 1)(a + 1)§ 21.

Формула разности квадратов351. 1) 4a 2 = (2a ) ; 9b2 = (3b) ; 16c2 = (4c) ; 0,04 x 2 = (0,2 x )2222)1 2 2 ⎛1 ⎞a b = ⎜ ab⎟ ;⎝3 ⎠9()2(20,81n6 = 0,9n3)25 6 45x z = ( x3z 2 )2 ;497)229 2 4 ⎛ 3 2⎞x y = ⎜ xy ⎟ ;⎝4⎠1623) 0,01a 4b2 = 0,1a 2b ;19 4 6 25 4 65m n =m n = ( m2n3 )216164352. 1) 25x 2 − 9 = (5x − 3) ⋅ (5x + 3) ;2) 4a 2 − 9 = (2a − 3) ⋅ (2a + 3)3) 64 y 2 − 36 x 2 = (8 y − 6 x ) ⋅ (8 y + 6 x ) ;4) 81a 2 − 16b2 = (9a − 4b) ⋅ (9a + 4b)1 2 16 2 1414y −x − ( y − x) ⋅ ( y + x)92535354121212) a 2 − b 2 = ( a − b) ⋅ ( a + b)9163434353. 1)3) 0,25a 2 − 0,49b2 = (0,5a − 0,7b) ⋅ (0,5a + 0,7b)4) 0,09 x 2 − 0,16 y 2 = (0,3x − 0,4 y ) ⋅ (0,3x + 0,4 y )354.

1) 36 x 2 y 2 − 1 = (6 xy − 1) ⋅ (6 xy + 1)2) x 2 y 4 − 16 = ( xy 2 − 4) ⋅ ( xy 2 + 4)3) 81a 6 − 49b 4 = (9a 3 − 7b 2 ) ⋅ (9a 3 + 7b 2 )4) 25a 2 − 9b 6 = (5a − 3b 3 ) ⋅ (5a + 3b 3 )7220,25x 2 y 2 = (0,5xy ) ;0,16m4 = 0,4m2 ;(2355. 1) a 4 − b 4 = (a 2 − b 2 ) ⋅ (a 2 + b 2 ) = (a − b ) ⋅ (a + b ) ⋅ (a 2 + b 2 )2) a 4 − b8 = (a 2 − b 4 ) ⋅ (a 2 + b 4 ) = (a − b 2 ) ⋅ (a + b 2 ) ⋅ (a 2 + b 2 )3) a 4 − 16 = (a 2 − 4) ⋅ (a 2 + 4) = (a − 2) ⋅ (a + 2) ⋅ (a 2 + 4)4) b4 − 81 = (b 2 − 9) ⋅ (b2 + 9) = (b − 3) ⋅ (b + 3) ⋅ (b2 + 4)356. 1) (2b + a ) ⋅ (2b − a ) = 4b2 − a 2 ; 2) (c + 3d ) ⋅ (c − 3d ) = c2 − 9d 23) ( y + 6 x ) ⋅ (6 x − y ) = 36 x 2 − y 2 ; 4) (3m − 2n) ⋅ (2n + 3m) = 9m2 − 4n2(3) (x)(− y )⋅ (y)()( ); 4) ( m − n ) ⋅ ( m + n ) = m357. 1) c2 + d 2 ⋅ c2 − d 2 = c4 − d 4 ; 2) a 2 + b3 ⋅ a 2 − b3 = a 4 − b6433)+ x4 = x2 − y6333()()2) (2m − 5n )⋅ (5n + 2m ) = 4m − 25n3) ( 0,2t + 0,5 p ) ⋅ ( 0,5 p − 0,2t ) = 0,25 p − 0,04t4) (1,2a − 0,3b ) ⋅ (1,2a + 0,3 p ) = 1,44a − 0,09b36− n6358.

1) 3a 2 + 4b3 ⋅ 3a 2 − 4b3 = 9a 4 − 16b6422484344386222244359. 1) 48 ⋅ 52 = (50 − 2) ⋅ (50 + 2) = 2500 − 4 = 24962) 68 ⋅ 72 = (70 + 2) ⋅ (70 − 2) = 4900 − 4 = 48963) 43 ⋅ 37 = (40 + 3) ⋅ (40 − 3) = 1600 − 9 = 15914) 47 ⋅ 53 = (50 − 3) ⋅ (50 + 3) = 2500 − 9 = 2491360. 1) 47 ⋅ 33 = (40 + 7) ⋅ (40 − 7) = 1600 − 49 = 15512) 44 ⋅ 36 = (40 + 4) ⋅ (40 − 4) = 1600 − 16 = 15843) 84 ⋅ 76 = (80 + 4) ⋅ (80 − 4) = 6400 − 16 = 63844) 201 ⋅ 199 = (200 + 1) ⋅ (200 − 1) = 40000 − 1 = 39999361. 1) (a + b) − c2 = (a + b + c) ⋅ (a + b − c)22) (m − n) − k 2 = (m − n − k ) ⋅ (m − n + k )23) (a + 2b) − 9a 2 = (a + 2b + 3a ) ⋅ (a + 2b − 3a ) = 4 ⋅ (2a + b) ⋅ (b − a )24) (3 x − y )2 − 4 y 2 = (3 x − y + 2 y ) ⋅ (3x − y − 2 y ) = 3 ⋅ (3 x + y ) ⋅ (x − y )73362.

1) (a − b )2 − (a − c )2 = (a − b − a + c ) ⋅ (a − b + a − c ) == (c − b ) ⋅ (2a − b − c )2) (a + b )2 − (b + c )2 = (a + b + b + c ) ⋅ (a + b − b − c ) == (a + 2b + c ) ⋅ (a − c )3) (2a + b) − (2b + a ) = (2a + b − 2b − a ) ⋅ (2a + b + 2b + a ) =22= 3 ⋅ (a − b) ⋅ (a + b)4) (a + 3b )2 − (3a + b )2 = (a + 3b − 3a − b ) ⋅ (a + 3b + 3a + b ) == (2b − 2a ) ⋅ (4a + 4b) = 8 ⋅ (b − a ) ⋅ (a + b)363. 1) 472 − 372 = (47 + 37) ⋅ (47 − 37) = 84 ⋅ 10 = 8402) 542 − 442 = (54 − 44) ⋅ (54 + 44) = 10 ⋅ 98 = 9803) 50,72 − 50,62 = (50,7 − 50,6) ⋅ (50,7 + 50,6) = 0,1 ⋅ 101,3 = 10,134) 29 ,42 − 29 ,32 = (29 ,4 − 29 ,3) ⋅ (29 ,4 + 29 ,3) = 0,1 ⋅ 58,7 = 5,87⎛ 2⎞⎝ 3⎠2⎛ 5⎞⎝ 9⎠2⎛ 1⎞⎝ 3⎠2⎛ 2⎝ 31⎞ ⎛ 23⎠ ⎝ 31⎞3⎠135) ⎜ 6 ⎟ − ⎜ 5 ⎟ = ⎜ 6 − 5 ⎟ ⋅ ⎜ 6 + 5 ⎟ = 1 ⋅ 12 = 16⎛ 4⎞⎝ 9⎠2⎛ 5⎝ 94⎞ ⎛ 59⎠ ⎝ 94⎞9⎠196) ⎜ 7 ⎟ − ⎜ 4 ⎟ = ⎜ 7 − 4 ⎟ ⋅ ⎜ 7 + 4 ⎟ = 3 ⋅ 12 = 37364.

1) ( x − 1) ⋅ ( x + 1) = x 2 − 2 ⋅ ( x − 3)x2 − 1 − x2 + 2 x − 6 = 02 x = 7 ; x = 3,52) 3 ⋅ ( x + 5) − x 2 = (2 − x) ⋅ (2 + x)3x + 15 – x2 = – x2 + 43x = – 11233) (2 x + 3) ⋅ (2 x + 3) − 4 ⋅ ( x − 1) ⋅ ( x + 1) = 49x = −34x2 + 12x + 9 – 4x2 + 4 = 4912x = 36; x = 34) (3 x + 1) ⋅ (3 x + 1) − (3 x − 2) ⋅ ( 2 + 3x ) = 179x2 + 3x + 3x + 1 – 9x2 + 4 = 176x + 5 = 17x=27413365. 1) (3 + x) ⋅ (3 − x) ⋅ (9 + x 2 ) = (9 − x 2 ) ⋅ (9 + x 2 ) = 81 − x 42) (4 x 2 + y 2 ) ⋅ (2 x + y ) ⋅ (2 x − y ) = (4 x 2 + y 2 ) ⋅ (4 x 2 − y 2 ) == 16 x 4 − y 4 .

(опечатка в ответе задачника).3) ( x 2 + 1) ⋅ ( x + 1) ⋅ ( x − 1) = ( x 2 + 1) ⋅ ( x 2 − 1) = x 4 − 14) (3a − 2b) ⋅ (3a + 2b) ⋅ (9a 2 + 4b 2 ) = (9a 2 − 4b 2 ) ⋅ (9a 2 + 4b 2 ) == 81a 4 − 16b 4366. 1)492 − 212 (49 − 21) ⋅ (49 + 21) 28 ⋅ 70 2 ⋅ 35 35====57 2 − 152 (57 − 15) ⋅ (57 + 15) 42 ⋅ 72 3 ⋅ 36 542)632 − 27 2 (63 − 27) ⋅ (63 + 27) 36 ⋅ 90 1 ⋅ 15 5====782 − 302 (78 − 30) ⋅ (78 + 30) 48 ⋅108 8 ⋅ 3 83)40,7 2 − 40,6 222=(40,7 − 40,6) ⋅ (40,7 + 40,6)81,3 ⋅ 0,1==(32,3 − 5,2) ⋅ (32,3 + 5,2)37,5 ⋅ 27,132,3 − 5,28,1331===37,5 ⋅ 27,1 375 1254)51,32 − 11,32(51,3 − 11,3) ⋅ (51,3 + 11,3)==22(113,9 − 73,9) ⋅ (113,9 + 73,9)113,9 − 73,9=40 ⋅ 62,6626 1==40 ⋅ 187,8 1878 3367.

Пусть x – первое число, тогда следующее за ним x + 1.|(x + 1)2 – x2| = |(x + 1 – x) ⋅ (x + 1 + x)| = |2x + 1| – нечетное число.368. (7n + 1) 2 – (2n – 4) 2 = (7n + 1 – 2n + 4) ⋅ (7n + 1 + 2n – 4) == (5n + 5) ⋅ (9n – 3) = 15 ⋅ (n + 1) ⋅ (3n – 1) M 15,т. к. 15(n + 1)(3n – 1) : 15 = (n + 1)(3n – 1).369. 1) (a + b)3 – (a – b)3 – 8b3 == (a2 + 2ab + b2) ⋅ (a + b) – (a2 – 2b + b2) ⋅ (a – b) – 8b3 == a3 + 2a2b + ab2 + a2b + 2ab2 – a3 + 2a2b – ab2 + a2b – 2ab2 ++ b3 – 8b3 = 6a 2b − 7b 3 = 6b ⋅ (a − b) ⋅ (a + b)2) (a 2 + b 2 ) 2 − (a 2 − b 2 ) 2 − a 2 == (a 2 + b 2 − a 2 + b 2 ) ⋅ ( a 2 + b 2 + a 2 − b 2 ) − a 2 = 2b 2 ⋅ 2a 2 − a 2 == a 2 ⋅ ( 4b 2 − 1) = a 2 ⋅ ( 2b − 1) ⋅ (2b + 1) .(опечатка в ответе задачника).753) (a 4 + b 4 ) 2 − (a 4 − b 4 ) 2 − a 2b 2 = 2b 2 ⋅ 2a 2 − a 2 == (a 4 + b 4 − a 4 + b 4 ) ⋅ (a 4 + b 4 + a 4 − b 4 ) − a 2b 2 = 2b 4 ⋅ 2a 4 − a 2b 2 == a 2b 2 ⋅ ( 2ab − 1) ⋅ (2ab + 1)4) 9a 4 − 13a 2b 2 + 4b 4 = 9a 4 − 9a 2b 2 − 4a 2b 2 + 4b 4 == 9a 2 ⋅ (a 2 − b 2 ) − 4b 2 ⋅ (a 2 − b 2 ) = (a 2 − b 2 ) ⋅ (9a 2 − 4b 2 ) == (a − b) ⋅ ( a + b) ⋅ (3a − 2b) ⋅ (3a + 2b)§ 22.

Квадрат суммы. Квадрат разности370. 1) (c + d ) 2 = c 2 + 2cd + d 2 ;3) (2 + x ) 2 = 4 + 4 x + x 2 ;2) ( x − y )2 = x 2 − 2 xy + y 24) ( x + 1)2 = x 2 + 2 x + 1371. 1) (q + 2 p)2 = q 2 + 4qp + 4 p 2 ;2) (3x + 2 y )2 = 9 x 2 + 12 xy + 4 y 23) (6a − 4b)2 = 36a 2 − 48ab + 16b 2 ;4) (5 z − t )2 = 25z 2 − 10 zt + t 2372. 1) (0,2 x + 0,3 y ) 2 = 0,04 x 2 + 0,12 xy + 0,09 y 22) (0,4b − 0,5c)2 = 0,16b 2 − 0,4bc + 0,25c 223⎞49⎛2;3) ⎜ x3 − ⎟ = x 6 − x3 +4⎠916⎝3⎛1⎝44⎞5⎠24) ⎜ a 3 − ⎟ =1 6 2 3 16a − a +16525373. 1) (−4ab − 5a 2 ) 2 = 16a 2b 2 + 40a 3b + 25a 42) (−3b 2 − 2ab) 2 = 9b 4 + 12ab 3 + 4a 2b 23) (0,2 x 2 + 5 xy) 2 = 0,04 x 4 + 2 x3 y + 25 x 2 y 24) (4 xy + 0,5 y 2 ) 2 = 16 x 2 y 2 + 4 xy3 + 0,25 y 4374. 1) (90 − 1)2 = 902 − 2 ⋅ 90 + 1 = 8100 − 180 + 1 = 79212) (40 + 1)2 = 402 + 2 ⋅ 40 + 1 = 1600 + 80 + 1 = 16813) 1012 = (100 + 1) 2 = 10000 + 200 + 1 = 102014) 98 2 = (100 − 2) 2 = 1000 − 400 + 4 = 9604375.

1) 72 2 = (70 + 2) 2 = 4900 + 280 + 4 = 51842) 57 2 = (60 − 3) 2 = 3600 − 360 + 9 = 3249763) 997 2 = (1000 − 3) 2 = 1000000 − 6000 + 9 = 9940094) 10012 = (1000 + 1) 2 = 1000000 + 2000 + 1 = 1002001376. (a + 1) 2 ≈ 1 + 2a1) 1,0052 = (1 + 0,005)2 ≈ 1 + 2 ⋅ 0,005 = 1,012) 1,004 2 = (1 + 0,004) 2 ≈ 1 + 2 ⋅ 0,004 = 1,0083) 1,012 2 = (1 + 0,012) 2 ≈ 1 + 2 ⋅ 0,012 = 1,0244) 1,0112 = (1 + 0,011) 2 ≈ 1 + 2 ⋅ 0,011 = 1,0225) 0,992 2 = (1 − 0,008) 2 ≈ 1 − 2 ⋅ 0,008 = 0,9846) 0,994 2 = (1 − 0,006) 2 ≈ 1 − 2 ⋅ 0,006 = 0,9887) 0,9882 = (1 − 0,012) 2 ≈ 1 − 2 ⋅ 0,012 = 0,9768) 0,989 2 = (1 − 0,011) 2 ≈ 1 − 2 ⋅ 0,011 = 0,978377.

1) a 2 + 4a + x = a 2 + 4a + 4 = ( a + 2) 2112) p 2 − 0,5 p + x = p 2 − 0,5 p + = ( p − ) 24163) 36a 2 − x + 49b 2 = 36a 2 − 84ab + 49b 2 = (6a − 7b) 24) a 2 − 6ab + x = a 2 − 6ab + 9b 2 = (a − 3b) 2378. 1) m 4 − 3m 2 + x = m 4 − 3m 2 + 2,25 = ( m 2 − 1,5) 22) a 2 + ab + x = a 2 + ab +b2 ⎛b⎞= ⎜a + ⎟4 ⎝2⎠3) 4a 2 − 5a + x = 4a 2 − 5a +25⎞25 ⎛= ⎜ 2a − ⎟4⎠16 ⎝24) x + 6a + 9a 2 = 1 + 6a + 9a 2 = (1 + 3a ) 2379. 1) 9a 2 − 6a + 1 = (3a − 1) 2 ;3) 36b 2 + 12b 2 + 1 = (6b + 1) 2 ;2) 1 + 2c + c 2 = (1 + c) 24) 81 − 18 x + x 2 = (9 − x) 2380.

1) 9 x 2 + 24 x + 16 = (3x + 4) 2 ;2) 100 − 60a + 9a 2 = (10 − 3a) 23) 36m 2 + 12nm + n 2 = (6m + n) 2 ;4) a 2 + 10ab + 25b 2 = (a + 5b) 277381. 1) x 4 + 2 x 2 y + y 2 = ( x 2 + y ) 2 ;2) p 4 − 2 p 2 q + q 2 = ( p 2 − q ) 23) 4c 4 + 12c 2 d 3 + 9d 6 = (2c 2 + 3d 3 ) 24) 25a 6 + 30a 3b + 9b 2 = (5a 3 + 3b) 2382. 1) a 4 − 8a 2 + 16 = (a 2 − 4) 2 = ( a − 2) 2 ⋅ (a + 2) 22) b 4 − 18b 2 + 81 = (b 2 − 9) 2 = (b − 3) 2 ⋅ (b + 3) 23) 25a 4 − 10a 2b + b 2 = (5a 2 − b) 24) 16 − 8a 2b 2 + a 4b 4 = (4 − a 2b 2 ) = ( 2 − ab) 2 ⋅ ( 2 + ab) 2383. 1) − a 2 − 2a − 1 = −(a + 1) 2 ;2) − 9 + 6b − b 2 = −(3 − b) 23) − 2a 2 + 8ab − 8b 2 = −2 ⋅ (a − 2b) 24) − 12ab − 3a 2 − 12b 2 = −3 ⋅ (a + 2b) 2384. 1) 16 x 2 − ( 4 x − 5) 2 = 15 ;2) 64 x 2 − (3 − 8 x) 2 = 8716 x 2 − 16 x 2 + 40 x − 25 = 15 ;40 x = 40 ;x =1;64 x 2 − 9 + 48 x − 64 x 2 = 8748 x = 96x=23) − 5 x ⋅ ( x − 3) + 5 ⋅ ( x − 1) 2 = −20− 5 x 2 + 15 x + 5 x 2 − 10 x + 5 = −205 x = −25 4 x = −54) (2 x − 3) 2 − ( 2 x + 3) 2 = 124 x 2 − 12 x + 9 − 4 x 2 − 12 x − 9 = 12124 x = −12 ; x = −2385.

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