Ejercicio 266

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CAPITULO XXXIII

Ecuaciones de segundo grado
Ejercicio 266
Resolver las siguientes ecuaciones llevándolas a la forma a x 2 +bx+c=0 y aplicando la fórmula general:
x= b ± b 2 4ac 2a
  1. x(x+3 ) =5x+3 x 2 +3x =5x+3 x 2 +3x5x3 =0 x 2 2x3 =0 x = 2 ± 2 2 4( 1 ) (3 ) 2( 1 ) x = 2 ± 4+12 2 x = 2 ± 16 2 x = 2 ± 4 2 x 1 = 2 x 1 =3 x 2 = 2 2 x 2 =1
  2. 3(3x2 ) =(x+4 ) (4x ) 9x6 = 4x x 2 +16 4x 9x616+ x 2 =0 x 2 +9x22 =0 x = 9 ± 9 2 4( 1 ) (22 ) 2( 1 ) x = 9 ± 81+88 2 x = 9 ± 169 2 x = 3 ± 13 2 x 1 = 2 x 1 =5 x 2 = 2 x 2 =4
  3. 9x+1 =3( x 2 5 ) (x3 ) (x+2 ) 9x+1 =3 x 2 15( x 2 x6 ) 9x+1 =3 x 2 15 x 2 +x+6 0 =2 x 2 9+x9x1 0 =2 x 2 8x10 Dividiendo la ecuación para 2 x 2 4x5 =0 x = 4 ± 4 2 4( 1 ) (5 ) 2( 1 ) x = 4 ± 16+20 2 x = 4 ± 36 2 x = 4 ± 6 2 x 1 = 2 2 x 1 =1 x 2 = 2 x 2 =5
  4. (2x3 ) 2 (x+5 ) 2 =23 4 x 2 12x+9( x 2 +10x+25 ) =23 4 x 2 12x+9 x 2 10x25 =23 3 x 2 22x16+23 =0 3 x 2 22x+7 =0 x = 22 ± 2 2 2 4( 3 ) ( 7 ) 2( 3 ) x = 22 ± 48484 6 x = 22 ± 400 6 x = 22 ± 20 6 x 1 = 2 x 1 = 1 3 x 2 = 6 x 2 =7
  5. 25 (x+2 ) 2 = (x7 ) 2 81 25( x 2 +4x+4 ) = x 2 14x+4981 25 x 2 +100x+100 = x 2 14x32 25 x 2 +100x+100 x 2 +14x+32 =0 24 x 2 +114x+132 =0 Dividiendo la ecuación para 6 4 x 2 +19x+22 =0 x = 19 ± 1 9 2 4( 4 ) ( 22 ) 2( 4 ) x = 19 ± 361352 8 x = 19 ± 9 8 x = 19 ± 3 8 x 1 = x 1 = 11 4 x 2 = 8 x 2 =2
  6. 3x(x2 ) (x6 ) =23(x3 ) 3 x 2 6xx+6 =23x69 3 x 2 7x+623x+69 =0 3 x 2 30x+75 =0 Dividiendo la ecuación para 3 x 2 10x+25 =0 x = 10 ± 1 0 2 4( 1 ) ( 25 ) 2( 1 ) x = 10 ± 2 x = 2 x =5
  7. 7(x3 ) 5( x 2 1 ) = x 2 5(x+2 ) 7x215 x 2 +5 = x 2 5x10 7x165 x 2 x 2 +5x+10 =0 6 x 2 +12x6 =0 Dividiendo para 6 la ecuación x 2 2x+1 =0 x = 2 ± 2 2 4( 1 ) ( 1 ) 2( 1 ) x = 2 ± 2 x = 2 2 x =1
  8. (x5 ) 2 (x6 ) 2 = (2x3 ) 2 118 x 2 10x+25( x 2 12x+36 ) =4 x 2 12x+9118 x 2 10x+25 x 2 +12x36 =4 x 2 12x109 0 =4 x 2 12x1092x+11 0 =4 x 2 14x98 Dividimos la ecuación para 2 2 x 2 7x49 =0 x = 7 ± 7 2 4( 2 ) (49 ) 2( 2 ) x = 7 ± 49+392 4 x = 7 ± 441 4 x = 7 ± 21 4 x 1 = 4 x 1 =7 x 2 = x 2 = 7 2
  9. (5x2 ) 2 (3x+1 ) 2 x 2 60 =0 25 x 2 20x+4(9 x 2 +6x+1 ) x 2 60 =0 25 x 2 20x9 x 2 6x1 x 2 56 =0 15 x 2 26x57 =0 x = 26 ± 2 6 2 4( 15 ) (57 ) 2( 15 ) x = 26 ± 676+3420 30 x = 26 ± 4096 30 x = 26 ± 64 30 x 1 = 30 x 1 =3 x 2 = x 2 = 19 15
  10. (x+4 ) 3 (x3 ) 3 =343 x 3 +12 x 2 +48x+64( x 3 9 x 2 +27x27 ) =343 x 3 +12 x 2 +48x+64 x 3 +9 x 2 27x+27343 =0 21 x 2 +21x252 =0 Dividiendo la ecuación para 21 x 2 +x12 =0 x = 1 ± 14( 1 ) (12 ) 2( 1 ) x = 1 ± 1+48 2 x = 1 ± 49 2 x = 1 ± 7 2 x 1 = 2 x 1 =4 x 2 = 2 x 2 =3
  11. (x+2 ) 3 (x1 ) 3 =x(3x+4 ) +8 x 3 +6 x 2 +12x+8( x 3 3 x 2 +3x1 ) =3 x 2 +4x+8 x 3 +6 x 2 +12x+ 8 x 3 + 3 x 2 3x+1 = 3 x 2 +4x+ 8 6 x 2 +9x+14x =0 6 x 2 +5x+1 =0 x = 5 ± 5 2 4( 6 ) ( 1 ) 2( 6 ) x = 5 ± 12 x = 5 ± 1 12 x 1 = 4 x 1 = 1 3 x 2 = 6 x 2 = 1 2
  12. (5x4 ) 2 (3x+5 ) (2x1 ) =20x(x2 ) +27 25 x 2 40x +16(6 x 2 3x+10x5 ) =20 x 2 40x +27 25 x 2 +166 x 2 7x+520 x 2 27 =0 x 2 7x6 =0 x 2 +7x+6 =0 x = 7 ± 7 2 4( 1 ) ( 6 ) 2( 1 ) x = 7 ± 4924 2 x = 7 ± 25 2 x = 7 ± 5 2 x 1 = 2 2 x 1 =1 x 2 = 2 x 2 =6