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CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 208
Elevar al cuadrado
  1. x 2 2x+1 = ([ x 2 2x ] +1 ) 2 = ( x 2 2x ) 2 +2( x 2 2x ) +1 = x 4 4 x 3 +4 x 2 +2 x 2 4x+1 = x 4 4 x 3 +6 x 2 4x+1
  2. 2 x 2 +x+1 = ([2 x 2 +x ] +1 ) 2 = (2 x 2 +x ) 2 +2(2 x 2 +x ) +1 =4 x 4 +4 x 3 + x 2 +4 x 2 +2x+1 =4 x 4 +4 x 3 +5 x 2 +2x+1
  3. x 2 5x+2 = ([ x 2 5x ] +2 ) 2 = ( x 2 5x ) 2 +4( x 2 5x ) + 2 2 = x 4 10 x 3 +25 x 2 +4 x 2 20x+4 = x 4 10 x 3 +29 x 2 20x+4
  4. x 3 5 x 2 +6 = ([ x 3 5 x 2 ] +6 ) 2 = ( x 3 5 x 2 ) 2 +12( x 3 5 x 2 ) + 6 2 = x 6 10 x 5 +25 x 4 +12 x 3 60 x 2 +36
  5. 4 a 4 3 a 2 +5 = ([4 a 4 3 a 2 ] +5 ) 2 = (4 a 4 3 a 2 ) 2 +10(4 a 4 3 a 2 ) + 5 2 =16 a 8 24 a 6 +9 a 4 +40 a 4 30 a 2 +25 =16 a 8 24 a 6 +49 a 4 30 a 2 +25
  6. x+2yz = ([x+2y ] z ) 2 = (x+2y ) 2 2z(x+2y ) + z 2 = x 2 +4xy+4 y 2 2xz4yz+ z 2
  7. 3 x 3 x 6 = ([3 x 3 ] x 6 ) 2 = (3 x 3 ) 2 2 x 6 (3 x 3 ) + x 12 =96 x 3 + x 6 6 x 6 +2 x 9 + x 12 =96 x 3 5 x 6 +2 x 9 + x 12
  8. 5 x 4 7 x 2 +3x = ([5 x 4 7 x 2 ] +3x ) 2 = (5 x 4 7 x 2 ) 2 +6x(5 x 4 7 x 2 ) + (3x ) 2 =25 x 8 70 x 6 +49 x 4 +30 x 5 42 x 3 +9 x 2 =25 x 8 70 x 6 +30 x 5 +49 x 4 42 x 3 +9 x 2
  9. 2 a 2 +2ab3 b 2 = ([2 a 2 +2ab ] 3 b 2 ) 2 = (2 a 2 +2ab ) 2 6 b 2 (2 a 2 +2ab ) + (3 b 2 ) 2 =4 a 4 +8 a 3 b+4 a 2 b 2 12 a 2 b 2 12a b 3 +9 b 4 =4 a 4 +8 a 3 b8 a 2 b 2 12a b 3 +9 b 4
  10. m 3 2 m 2 n+2 n 4 = ([ m 3 2 m 2 n ] +2 n 4 ) 2 = ( m 3 2 m 2 n ) 2 +4 n 4 ( m 3 2 m 2 n ) + (2 n 4 ) 2 = m 6 4 m 5 n+4 m 4 n 2 +4 m 3 n 4 8 m 2 n 5 +4 n 8
  11. a 2 b+ c 4 = ([ a 2 b ] + c 4 ) 2 = ( a 2 b ) 2 + c 2 ( a 2 b ) + ( c 4 ) 2 = a 2 4 ab+ b 2 + ac 4 bc 2 + c 2 16 = a 2 4 + b 2 + c 2 16 ab+ ac 4 bc 2
  12. x 5 5y+ 5 3 = ([ x 5 5y ] + 5 3 ) 2 = ( x 5 5y ) 2 + 10 3 ( x 5 5y ) + ( 5 3 ) 2 = x 2 25 2xy+25 y 2 + 2 3 x 50 3 y+ 25 9
  13. 1 2 x 2 x+ 2 3 = ([ 1 2 x 2 x ] + 2 3 ) 2 = ( 1 2 x 2 x ) 2 + 4 3 ( 1 2 x 2 x ) + ( 2 3 ) 2 = x 4 4 x 3 + x 2 + 2 3 x 2 4 3 x+ 4 9 = x 4 4 x 3 + 5 3 x 2 4 3 x+ 4 9
  14. a x 1 3 + x a = ([ a x 1 3 ] + x a ) 2 = ( a x 1 3 ) 2 + 2x a ( a x 1 3 ) + ( x a ) 2 = a 2 x 2 2a 3x + 1 9 +2 2x 3a + x 2 a 2 = a 2 x 2 2a 3x 2x 3a + x 2 a 2 + 19 9
  15. 3 4 a 2 1 2 a+ 4 5 = ([ 3 4 a 2 1 2 a ] + 4 5 ) 2 = ( 3 4 a 2 1 2 a ) 2 + 8 3 ( 3 4 a 2 1 2 a ) + ( 4 5 ) 2 = 9 16 a 4 3 4 a 3 + 1 4 a 2 +2 a 2 4 3 a+ 16 25 = 9 16 a 4 3 4 a 3 + 29 20 a 2 4 3 a+ 16 25
  16. a 2 4 3 5 + b 2 9 = ([ a 2 4 3 5 ] + b 2 9 ) 2 = ( a 2 4 3 5 ) 2 + 2 b 2 9 ( a 2 4 3 5 ) + ( b 2 9 ) 2 = a 4 16 3 10 a 2 + 9 25 + a 2 b 2 18 2 b 2 15 + b 4 81 = a 4 16 3 10 a 2 + a 2 b 2 18 2 b 2 15 + b 4 81 + 9 25
  17. x 3 x 2 +x+1 = ([ x 3 x 2 ] +[x+1 ] ) 2 = ( x 3 x 2 ) 2 +2( x 3 x 2 ) (x+1 ) + (x+1 ) 2 = x 6 2 x 5 + x 4 +2( x 4 + x 3 x 3 x 2 ) + x 2 +2x+1 = x 6 2 x 5 + x 4 +2 x 4 2 x 2 + x 2 +2x+1 = x 6 2 x 5 +3 x 4 x 2 +2x+1
  18. x 3 3 x 2 2x+2 = ([ x 3 3 x 2 ] [2x2 ] ) 2 = ( x 3 3 x 2 ) 2 2( x 3 3 x 2 ) (2x2 ) + (2x2 ) 2 = x 6 6 x 5 +9 x 4 2(2 x 4 2 x 3 6 x 3 +6 x 2 ) +4 x 2 8x+4 = x 6 6 x 5 +9 x 4 4 x 4 +16 x 3 12 x 2 +4 x 2 8x+4 = x 6 6 x 5 +5 x 4 +16 x 3 8 x 2 8x+4
  19. x 4 +3 x 2 4x+5 = ([ x 4 +3 x 2 ] [4x5 ] ) 2 = ( x 4 +3 x 2 ) 2 2( x 4 +3 x 2 ) (4x5 ) + (4x5 ) 2 = x 8 +6 x 6 +9 x 4 2(4 x 5 5 x 4 +12 x 3 15 x 2 ) +16 x 2 40x+25 = x 8 +6 x 6 +9 x 4 8 x 5 +10 x 4 24 x 3 +30 x 2 +16 x 2 40x+25 = x 8 +6 x 6 8 x 5 +19 x 4 24 x 3 +46 x 2 40x+25
  20. x 4 4 x 3 +2x3 = ([ x 4 4 x 3 ] +[2x3 ] ) 2 = ( x 4 4 x 3 ) 2 +2( x 4 4 x 3 ) (2x3 ) + (2x3 ) 2 = x 8 8 x 7 +16 x 6 +2(2 x 5 3 x 4 8 x 4 +12 x 3 ) +4 x 2 12x+9 = x 8 8 x 7 +16 x 6 +4 x 5 22 x 4 +24 x 3 +4 x 2 12x+9
  21. 36a+ a 2 a 3 = ([36a ] +[ a 2 a 3 ] ) 2 = (36a ) 2 +2(36a ) ( a 2 a 3 ) + ( a 2 a 3 ) 2 =936a+36 a 2 +2(3 a 2 3 a 3 6 a 3 +6 a 4 ) + a 4 2 a 5 + a 6 =936a+36 a 2 +6 a 2 18 a 3 +12 a 4 + a 4 2 a 5 + a 6 =936a+42 a 2 18 a 3 +13 a 4 2 a 5 + a 6
  22. 1 2 x 3 x 2 + 2 3 x+2 = ([ 1 2 x 3 x 2 ] +[ 2 3 x+2 ] ) 2 = ( 1 2 x 3 x 2 ) 2 +2( 1 2 x 3 x 2 ) ( 2 3 x+2 ) + ( 2 3 x+2 ) 2 = x 6 4 x 5 + x 4 +2( 1 3 x 4 + x 3 2 3 x 3 2 x 2 ) + 4 9 x 2 + 8 3 x+4 = x 6 4 x 5 + x 4 + 2 3 x 4 + 2 3 x 3 4 x 2 + 4 9 x 2 + 8 3 x+4 = x 6 4 x 5 + 5 3 x 4 + 2 3 x 3 32 9 x 2 + 8 3 x+4
  23. 1 2 a 3 2 3 a 2 + 3 4 a 1 2 = ([ 1 2 a 3 2 3 a 2 ] +[ 3 4 a 1 2 ] ) 2 = ( 1 2 a 3 2 3 a 2 ) 2 +2( 1 2 a 3 2 3 a 2 ) ( 3 4 a 1 2 ) + ( 3 4 a 1 2 ) 2 = a 6 4 2 a 5 3 + 4 a 4 9 +2( 3 a 4 8 a 3 4 a 3 2 + a 2 3 ) + 9 a 2 16 3a 4 + 1 4 = a 6 4 2 a 5 3 + 4 a 4 9 + 3 a 4 4 3 a 3 2 + 2 a 2 3 + 9 a 2 16 3a 4 + 1 4 = a 6 4 2 a 5 3 + 43 a 4 36 3 a 3 2 + 59 a 2 48 3a 4 + 1 4
  24. x 5 x 4 + x 3 x 2 +x2 = ([ x 5 x 4 + x 3 ] [ x 2 x+2 ] ) 2 = ( x 5 x 4 + x 3 ) 2 2( x 5 x 4 + x 3 ) ( x 2 x+2 ) + ( x 2 x+2 ) 2 = ([ x 5 x 4 ] + x 3 ) 2 2( x 7 x 6 +2 x 5 x 6 + x 5 2 x 4 + x 5 x 4 +2 x 3 ) + ([ x 2 x ] +2 ) 2 = ( x 5 x 4 ) 2 +2 x 3 ( x 5 x 4 ) + x 6 2( x 7 2 x 6 +4 x 5 3 x 4 +2 x 3 ) + ( x 2 x ) 2 +4( x 2 x ) + 2 2 = x 10 2 x 9 + x 8 +2 x 8 2 x 7 + x 6 2 x 7 +4 x 6 8 x 5 +6 x 4 4 x 3 + x 4 2 x 3 + x 2 +4 x 2 4x+4 = x 10 2 x 9 +3 x 8 4 x 7 +5 x 6 8 x 5 +7 x 4 6 x 3 +5 x 2 4x+4