Ejercicio 268

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CAPITULO XXXIII

Ecuaciones de segundo grado con denominadores
Ejercicio 268
Resolver las siguientes ecuaciones:
  1. x 2 5 x 2 = 3 10 x 2 5 x 2 = 3 10 2 x 2 5x 10 = 3 10 2 x 2 5x3 =0 x= 5 ± (5 ) 2 4( 2 ) (3 ) 2( 2 ) x= 5 ± 25+24 4 x= 5 ± 49 4 x= 5 ± 7 4 x 1 = 5+7 4 x 1 = 4 x 1 =3 x 2 = 57 4 x 2 = 2 x 2 = 1 2
  2. 4x 13 x = 3 2 4x 13 x 3 2 =0 8 x 2 263x 2x =0 8 x 2 3x26 =0 x= 3 ± (3 ) 2 4( 8 ) (26 ) 2( 8 ) x= 3 ± 9+832 16 x= 3 ± 841 16 x= 3 ± 29 16 x 1 = 3+29 16 x 1 = 16 x 1 =2 x 2 = 329 16 x 2 = x 2 = 13 8
  3. x 2 6 x 2 =3(x5 ) x 2 3x 6 =3(x5 ) x 2 3x =18(x5 ) x 2 3x =18x90 x 2 3x18x+90 =0 x 2 21x+90 =0 m=21 n=90 x= 21 2 ± (21 ) 2 4 90 x= 21 2 ± 441 4 90 x= 21 2 ± 441360 4 x= 21 2 ± 81 4 x= 21 2 ± 9 2 x 1 = 21 2 + 9 2 = 2 x 1 =15 x 2 = 21 2 9 2 = 2 x 2 =6
  4. 1 4 (x4 ) + 2 5 (x5 ) = 1 5 ( x 2 53 ) Multiplico la ecuación por 20 5(x4 ) +8(x5 ) =4( x 2 53 ) 5x20+8x40 =4 x 2 212 0 =4 x 2 21213x+60 4 x 2 13x152 =0 x= 13 ± (13 ) 2 4( 4 ) (152 ) 2( 4 ) x= 13 ± 169+2432 8 x= 13 ± 2601 8 x= 13 ± 51 8 x 1 = 13+51 8 x 1 = 8 x 1 =8 x 2 = 1351 8 x 2 = x 2 = 19 4
  5. 5 x 1 x+2 =1 5(x+2 ) x x(x+2 ) =1 5(x+2 ) x =x(x+2 ) 5x+10x = x 2 +2x 0 = x 2 +2x4x10 x 2 2x10 =0 m=2 n=10 x= 2 2 ± 2 2 4 (10 ) x=1 ± 1+10 x=1 ± 11 x 1 =1+ 11 x 2 =1 11
  6. 15 x 11x+5 x 2 =1 15x11x5 x 2 =1 4x5 = x 2 x 2 +4x5 =0 m=4 n=5 x= 2 ± 4 2 4 (5 ) x=2 ± 4+5 x=2 ± 9 x=2 ± 3 x 1 =2+3 x 1 =1 x 2 =23 x 2 =5
  7. 8x 3x+5 + 5x1 x+1 =3 8x(x+1 ) +(5x1 ) (3x+5 ) (3x+5 ) (x+1 ) =3 8x(x+1 ) +(5x1 ) (3x+5 ) =3(3x+5 ) (x+1 ) 8 x 2 +8x+15 x 2 +25x3x5 =3(3 x 2 +3x+5x+5 ) 23 x 2 +30x5 =9 x 2 +24x+15 23 x 2 +30x59 x 2 24x15 =0 14 x 2 +6x20 =0 Dividiendo la ecuación para 2 7 x 2 +3x10 =0 x= 3 ± 3 3 4( 7 ) (10 ) 2( 7 ) x= 3 ± 9+280 14 x= 3 ± 289 14 x= 3 ± 17 14 x 1 = 3+17 14 = 14 14 x 1 =1 x 2 = 317 14 = x 2 = 10 7
  8. 1 x2 1 x1 = 1 6 x 1 x +2 (x2 ) (x1 ) = 1 6 1 (x2 ) (x1 ) = 1 6 6 =(x2 ) (x1 ) 6 = x 2 3x+2 0 = x 2 3x+26 x 2 3x4 =0 m=3 n=4 x= 3 2 ± (3 ) 2 4 (4 ) x= 3 2 ± 9 4 +4 x= 3 2 ± 9+16 4 x= 3 2 ± 25 4 x= 3 2 ± 5 2 x 1 = 3 2 + 5 2 = 3+5 2 = 2 x 1 =4 x 2 = 3 2 5 2 = 35 2 = 2 2 x 2 =1
  9. 1 2x3 x+5 = x2 10 x+52x+3 x+5 = x2 10 8x x+5 = x2 10 10(8x ) =(x2 ) (x+5 ) 8010x = x 2 +3x10 0 = x 2 +3x1080+10x x 2 +13x90 =0 m=13 n=90 x= 13 2 ± (13 ) 2 4 (90 ) x= 13 2 ± 169 4 +90 x= 13 2 ± 169+360 4 x= 13 2 ± 529 4 x= 13 2 ± 23 2 x 1 = 13 2 + 23 2 = 13+23 2 = 2 x 1 =5 x 2 = 13 2 23 2 = 1323 2 = 2 x 2 =18
  10. x13 x =5 10(5x+3 ) x 2 x13 x = 5 x 2 10(5x+3 ) x 2 x(x13 ) =5 x 2 50x30 x 2 13x =5 x 2 50x30 0 =5 x 2 50x30 x 2 +13x 4 x 2 37x30 =0 x= 37 ± (37 ) 2 4( 4 ) (30 ) 2( 4 ) x= 37 ± 1369+480 8 x= 37 ± 1849 8 x= 37 ± 43 8 x 1 = 37+43 8 = 8 x 1 =10 x 2 = 3743 8 = x 2 = 3 4
  11. x x2 x2 x = 5 2 x 2 (x2 ) 2 x(x2 ) = 5 2 x 2 ( x 2 4x+4 ) x 2 2x = 5 2 x 2 x 2 +4x4 x 2 2x = 5 2 2(4x4 ) =5( x 2 2x ) 8x8 =5 x 2 10x 0 =5 x 2 10x8x+8 5 x 2 18x+8 =0 x= 18 ± (18 ) 2 4( 5 ) ( 8 ) 2( 5 ) x= 18 ± 324160 10 x= 18 ± 164 10 x= 18 ± 2 2 .41 10 x= 18 ± 2 41 10 x= 2 (9 ± 41 ) x 1 = 9+ 41 5 x 2 = 9 41 5
  12. 4 x 2 x1 13x 4 = 20x 3 16 x 2 (x1 ) (13x ) 4(x1 ) = 20x 3 16 x 2 (x3 x 2 1+3x ) 4x4 = 20x 3 16 x 2 +3 x 2 4x+1 4x4 = 20x 3 3(19 x 2 4x+1 ) =20x(4x4 ) 57 x 2 12x+3 =80 x 2 80x 0 =80 x 2 80x57 x 2 +12x3 23 x 2 68x3 =0 x= 68 ± (68 ) 2 4( 23 ) (3 ) 2( 23 ) x= 68 ± 4624+276 46 x= 68 ± 4900 46 x= 68 ± 70 46 x 1 = 68+70 46 = 46 x 1 =3 x 2 = 6870 46 = 2 x 2 = 1 23
  13. 3x1 x 2x 2x1 7 6 =0 6(2x1 ) (3x1 ) 12 x 2 7x(2x1 ) 6x(2x1 ) =0 6(2x1 ) (3x1 ) 12 x 2 7x(2x1 ) =0 6(6 x 2 2x3x+1 ) 12 x 2 14 x 2 +7x =0 36 x 2 30x+626 x 2 +7x =0 10 x 2 23x+6 =0 x= 23 ± (23 ) 2 4( 10 ) ( 6 ) 2( 10 ) x= 23 ± 529240 20 x= 23 ± 289 20 x= 23 ± 17 20 x 1 = 23+17 20 = 20 x 1 =2 x 2 = 2317 20 = x 2 = 3 10
  14. 5x8 x1 = 7x4 x+2 (5x8 ) (x+2 ) =(7x4 ) (x1 ) 5 x 2 +10x8x16 =7 x 2 7x4x+4 0 =7 x 2 11x+45 x 2 2x+16 2 x 2 13x+20 =0 x= 13 ± (13 ) 2 4( 2 ) ( 20 ) 2( 2 ) x= 13 ± 169160 4 x= 13 ± 9 4 x= 13 ± 3 4 x 1 = 13+3 4 = 4 x 1 =4 x 2 = 133 4 = x 2 = 5 2
  15. x+3 2x1 5x1 4x+7 =0 (x+3 ) (4x+7 ) (5x1 ) (2x1 ) (2x1 ) (4x+7 ) =0 (x+3 ) (4x+7 ) (5x1 ) (2x1 ) =0 4 x 2 +7x+12x+21(10 x 2 5x2x+1 ) =0 4 x 2 +19x+2110 x 2 +7x1 =0 6 x 2 +26x+20 =0 Dividiendo la ecuación para 2 3 x 2 13x10 =0 x= 13 ± (13 ) 2 4( 3 ) (10 ) 2( 3 ) x= 13 ± 169+120 6 x= 13 ± 289 6 x= 13 ± 17 6 x 1 = 13+17 6 = 6 x 1 =5 x 2 = 1317 6 = x 2 = 2 3
  16. 1 4x 1 6 = 1 x+1 64+x 6(4x ) = 1 x+1 2+x 246x = 1 x+1 (2+x ) (x+1 ) =246x x 2 +3x+2 =246x x 2 +3x+224+6x =0 x 2 +9x22 =0 m=9 n=22 x= 9 2 ± 9 2 4 (22 ) x= 9 2 ± 81 4 +22 x= 9 2 ± 81+88 4 x= 9 2 ± 169 4 x= 9 2 ± 13 2 x 1 = 9 2 + 13 2 = 2 x 1 =2 x 2 = 9 2 13 2 = 2 x 2 =11
  17. x+4 x+5 x+2 x+3 = 1 24 (x+4 ) (x+3 ) (x+2 ) (x+5 ) (x+5 ) (x+3 ) = 1 24 x 2 +7x+12( x 2 +7x+10 ) x 2 +8x+15 = 1 24 x 2 + 7x +12 x 2 7x 10 x 2 +8x+15 = 1 24 2 x 2 +8x+15 = 1 24 48 = x 2 +8x+15 0 = x 2 +8x+1548 x 2 +8x33 =0 m=8 n=33 x= 2 ± 8 2 4 (33 ) x=4 ± 4 +33 x=4 ± 16+33 x=4 ± 49 x=4 ± 7 x 1 =4+7 x 1 =3 x 2 =47 x 2 =11
  18. 5 x 2 1 6 x+1 =3 5 8 5 (x+1 ) (x1 ) 6 x+1 = 29 8 56(x1 ) (x+1 ) (x1 ) = 29 8 56x+6 x 2 1 = 29 8 116x x 2 1 = 29 8 8(116x ) =29( x 2 1 ) 8848x =29 x 2 29 0 =29 x 2 2988+48x 29 x 2 +48x117 =0 x= 48 ± 4 8 2 4( 29 ) (117 ) 2( 29 ) x= 48 ± 2304+13572 58 x= 48 ± 15876 58 x= 48 ± 126 58 x 1 = 48+126 58 x 1 = x 1 = 39 29 x 2 = 48126 58 x 2 = 58 x 2 =3
  19. x1 x+1 + x+1 x1 = 2x+9 x+3 (x1 ) 2 + (x+1 ) 2 (x+1 ) (x1 ) = 2x+9 x+3 (x1 ) 2 + (x+1 ) 2 x 2 1 = 2x+9 x+3 x 2 2x +1+ x 2 + 2x +1 x 2 1 = 2x+9 x+3 2 x 2 +2 x 2 1 = 2x+9 x+3 (2 x 2 +2 ) (x+3 ) =( x 2 1 ) (2x+9 ) 2 x 3 +2x+6 x 2 +6 = 2 x 3 +9 x 2 2x9 0 =9 x 2 2x92x6 x 2 6 3 x 2 4x15 =0 x= 4 ± (4 ) 2 4( 3 ) (15 ) 2( 3 ) x= 4 ± 16+180 6 x= 4 ± 196 6 x= 4 ± 14 6 x 1 = 4+14 6 x 1 = 6 x 1 =3 x 2 = 414 6 x 2 = x 2 = 5 3
  20. 3 x+2 1 x2 = 1 x+1 3(x2 ) (x+2 ) (x+2 ) (x2 ) = 1 x+1 3x6x2 x 2 4 = 1 x+1 2x8 x 2 4 = 1 x+1 (2x8 ) (x+1 ) = x 2 4 2 x 2 8x+2x8 = x 2 4 2 x 2 6x8 x 2 +4 =0 x 2 6x4 =0 m=6 n=4 x= 2 ± (6 ) 2 4 (4 ) x=3 ± 4 +4 x=3 ± 9+4 x=3 ± 13 x 1 =3+ 13 x 2 =3 13