Ejercicio 274

CAPITULO XXXIII

Representación y solución gráfica de ecuaciones de segundo grado
Para graficar este grupo de ejercicios, se utilizara el software matemático Geogebra 6, el cual es un programa multiplataforma y cada ejercicio tendrá un link para la descarga del mismo realizado en Geogebra 6.
Ejercicio 274
Representar graficamente las funciones:
  1. x 2 +3x4
    x
    y= x 2 +3x4
    -1
    6
    0
    -4
    1
    0
    2
    6

    ecuacion cuadratica
  2. x 2 +3x+2
    x
    y= x 2 +3x+2
    -1
    0
    0
    2
    1
    6
    2
    12

    ecuacion segundo grado
  3. x 2 5x+6
    x
    y= x 2 5x+6
    -1
    13
    0
    6
    1
    2
    2
    0

    ecuacion de segundo grado
  4. x 2 +2x8
    x
    y= x 2 +2x8
    -1
    -9
    0
    -8
    1
    -5
    2
    0

    grafico de ecuacion
  5. x 2 2x8
    x
    y= x 2 2x8
    -1
    -5
    0
    -8
    1
    -9
    2
    -8

    parabola
  6. x 2 9
    x
    y= x 2 9
    -3
    0
    0
    -9
    1
    -8
    3
    0

    ecuacion de segundo grado
  7. x 2 8x+16
    x
    y= x 2 8x+16
    -1
    5
    0
    16
    1
    9
    2
    4

    grafico ecuacion de segundo grado
  8. x 2 +4x+4
    x
    y= x 2 +4x+4
    -1
    1
    0
    4
    1
    9
    2
    14

    ecuacion de segundo grado
  9. 2 x 2 9x+7
    x
    y=2 x 2 9x+7
    -1
    18
    0
    7
    1
    0
    2
    -3

    ecuacion de segundo grado
  10. 3 x 2 4x7
    x
    y=3 x 2 4x7
    -1
    0
    0
    -7
    1
    -8
    2
    -3


    Resolver gráficamente las ecuaciones:
    3x2-4x-7
  11. x 2 4x+3=0
    x
    y= x 2 4x+3
    -1
    8
    0
    3
    1
    0
    2
    -1

    x2-4x+3=0
  12. x 2 6x+8=0
    x
    y= x 2 6x+8
    -1
    15
    0
    8
    1
    3
    2
    0

    x2-6x+8=0
  • x 2 2x3=0
    x
    y= x 2 2x3
    -1
    0
    0
    -3
    1
    -4
    2
    -3

    x2-6x+8=0
  • x 2 +4x+3=0
    x
    y= x 2 +4x+3
    -1
    0
    0
    3
    1
    8
    2
    15

    x2+4x+3=0
  • x 2 2x8=0
    x
    y= x 2 2x8
    -1
    -5
    0
    -8
    1
    -9
    2
    -8

    x2-2x-8=0
  • x 2 =2x1
    x
    y= x 2 2x+1
    -1
    4
    0
    1
    1
    0
    2
    1

    x2-2x-8=7
  • x 2 +8x+16=0
    x
    y= x 2 +8x+16
    -1
    9
    0
    16
    1
    25
    -2
    4

    x2+8x+16=0
  • x 2 4=0
    x
    y= x 2 4
    -2
    0
    1
    -3
    0
    -4
    2
    0

    x2+8x+16=10
  • x 2 =3x+10
    x
    y= x 2 3x10
    -1
    -6
    0
    -10
    1
    -12
    2
    -15

    x2=3x+10
  • x 2 4x=4
    x
    y= x 2 4x+4
    -1
    9
    0
    4
    1
    1
    2
    0

    x2-4x=-4
  • 2 x 2 9x+10=0
    x
    y=2 x 2 9x+10
    -1
    21
    0
    10
    1
    3
    2
    0

    x2-4x=-46

  • 2 x 2 5x7=0
    x
    y=2 x 2 5x7
    -1
    0
    0
    -7
    1
    -10
    2
    -9

    2x2-5x-7=0

    Ejercicio 273

    CAPITULO XXXIII

    Ecuaciones con radicales que se reducen a segundo grado (soluciones extrañas)
    Ejercicio 273
    Resolver las ecuaciones siguientes haciendo la verificación con ambas raíces:
    1. x+ 4x+1 =5 ( 4x+1 ) 2 = (5x ) 2 4x+1 =2510x+ x 2 0 =4x1+2510x+ x 2 x 2 14x+24 =0 (x12 ) (x2 ) =0 x2 =0 x 1 =2 x12 =0 x 2 =12 Verificación : Sea x=2 2+ 4( 2 ) +1 =5 2+ 9 =5 2+3 =5 5 =5x=2  si satisface la ecuación dada Sea x=12 12+ 4( 12 ) +1 =5 12+ 25 =5 12+5 =5 17 5x=12  no satisface la ecuación, por tanto es una solución extraña
    2. 2x x1 =3x7 x1 =3x2x7 ( x1 ) 2 = (x7 ) 2 x1 = x 2 14x+49 0 = x 2 14xx+1+49 x 2 15x+50 =0 (x10 ) (x5 ) =0 x5 =0 x 1 =5 x10 =0 x 2 =10 Verificación : Sea x=5 2( 5 ) 51 =3( 5 ) 7 10 4 =157 102 =8 8 =8x=5  si satisface la ecuación dada Sea x=10 2( 10 ) 101 =3( 10 ) 7 20 9 =307 203 =23 17 23x=10  no satisface la ecuación, por tanto es una solución extraña
    3. 5x1 + x+3 =4 ( 5x1 ) 2 = (4 x+3 ) 2 5x1 =168 x+3 +x+3 5x1x316 =8 x+3 4x20 =8 x+3 Dividiendo la ecuación para 4 (x5 ) 2 = (2 x+3 ) 2 x 2 10x+25 =4(x+3 ) x 2 10x+25 =4x+12 x 2 10x+254x12 =0 x 2 14x+13 =0 (x13 ) (x1 ) =0 x1 =0 x 1 =1 x13 =0 x 2 =13 Verificación : Sea x=1 5( 1 ) 1 + 1+3 =4 4 + 4 =4 2+2 =4 4 =4x=1  si satisface la ecuación dada Sea x=13 5( 13 ) 1 + 13+3 =4 64 + 16 =4 8+4 =4 12 4x=12  no satisface la ecuación, por tanto es una solución extraña
    4. 2 x x+5 =1 (2 x 1 ) 2 = ( x+5 ) 2 4x4 x +1 =x+5 4xx5+1 =4 x (3x4 ) 2 = (4 x ) 2 9 x 2 24x+16 =16x 9 x 2 24x+1616x =0 9 x 2 40x+16 =0 9 x 2 36x4x+16 =0 9x(x4 ) 4(x4 ) =0 (x4 ) (9x4 ) =0 x4 =0 x 1 =4 9x4 =0 9x =4 x 2 = 4 9 Verificación : Sea x=4 2 4 4+5 =1 2( 2 ) 9 =1 43 =1 1 =1x=4  si satisface la ecuación dada Sea x= 4 9 2 4 9 4 9 +5 =1 2( 2 3 ) 4+45 9 =1 4 3 49 9 =1 4 3 7 3 =1 47 3 =1 1 1x= 4 9  no satisface la ecuación, por tanto es una solución extraña
    5. 2x1 + x+3 =3 ( 2x1 ) 2 = (3 x+3 ) 2 2x1 =96 x+3 +x+3 2x112x =6 x+3 (x13 ) 2 = (6 x+3 ) 2 x 2 26x+169 =36(x+3 ) x 2 26x+169 =36x+108 x 2 26x+16936x108 =0 x 2 62x+61 =0 (x61 ) (x1 ) =0 x1 =0 x 1 =1 x61 =0 x 2 =61 Verificación : Sea x=1 2( 1 ) 1 + 1+3 =3 21 + 4 =3 1+2 =3 3 =3x=1  si satisface la ecuación dada Sea x=61 2( 61 ) 1 + 61+3 =3 1221 + 64 =3 121 +8 =3 11+8 =3 18 3x=61  no satisface la ecuación, por tanto es una solución extraña
    6. x3 + 2x+1 2 x =0 ( x3 + 2x+1 ) 2 = (2 x ) 2 x3+2 (x3 ) (2x+1 ) +2x+1 =4x 2 2 x 2 +x6x3 =4x3x+2 (2 2 x 2 5x3 ) 2 = (x+2 ) 2 4(2 x 2 5x3 ) = x 2 +4x+4 8 x 2 20x12 x 2 4x4 =0 7 x 2 24x16 =0 7 x 2 28x+4x16 =0 7x(x4 ) +4(x4 ) =0 (7x+4 ) (x4 ) =0 x4 =0 x 1 =4 7x+4 =0 7x =4 x 2 = 4 7 Verificación : Sea x=4 43 + 2( 4 ) +1 2 4 =0 1 + 8+1 2( 2 ) =0 1+34 =0 0 =0x=4  si satisface la ecuación dada Sea x= 4 7 4 7 3 + 2( 4 7 ) +1 2 4 7 =0 421 7 + 8 7 +1 2 4 7 =0 25 7 + 8+7 7 2 2 7 i =0 5 7 i + 1 7 i 2 2 7 i =0 5 7 i + 1 7 i 4 7 i 0x= 4 7  no satisface la ecuación, por tanto es una solución extraña
    7. 5x1 3x = 2x ( 5x1 3x ) 2 = ( 2x ) 2 5x12 (5x1 ) (3x ) +3x =2x 2 (5x1 ) (3x ) =2x4x2 2 15x5 x 2 3+x =2x2 Dividiendo la ecuación para 2 ( 16x5 x 2 3 ) 2 = (x+1 ) 2 16x5 x 2 3 = x 2 +2x+1 16x5 x 2 3 x 2 2x1 =0 6 x 2 +14x4 =0 Dividiendo la ecuación para 2 3 x 2 7x+2 =0 3 x 2 6xx+2 =0 3x(x2 ) (x2 ) =0 (3x1 ) (x2 ) =0 x2 =0 x 1 =2 3x1 =0 3x =1 x 2 = 1 3 Verificación : Sea x=2 5( 2 ) 1 32 = 2( 2 ) 101 1 = 2 2 9 1 =2 31 =2 2 =2x=2  si satisface la ecuación dada Sea x= 1 3 5( 1 3 ) 1 3 1 3 = 2( 1 3 ) 5 3 1 91 3 = 2 3 53 3 8 3 = 2 3 2 3 8 3 = 2 3 8 3 0x= 1 3  no satisface la ecuación, por tanto es una solución extraña
    8. 3x+1 + 5x = 16x+1 ( 3x+1 + 5x ) 2 = ( 16x+1 ) 2 3x+ 1 +2 5x(3x+1 ) +5x =16x+ 1 2 15 x 2 +5x =16x8x 2 15 x 2 +5x =8x Dividiendo la ecuación para 2 ( 15 x 2 +5x ) 2 = (4x ) 2 15 x 2 +5x =16 x 2 0 =16 x 2 15 x 2 5x x 2 5x =0 x(x5 ) =0 x 1 =0 x5 =0 x 2 =5 Verificación : Sea x=0 3( 0 ) +1 + 5( 0 ) = 16( 0 ) +1 1 +0 = 0+1 1 =1x=0  si satisface la ecuación dada Sea x=5 3( 5 ) +1 + 5( 5 ) = 16( 5 ) +1 15+1 + 5 2 = 80+1 16 +5 = 81 4+5 =9 9 =9x=5  si satisface la ecuación
    9. 2x+ 4x3 =3 ( 2x+ 4x3 ) 2 = 3 2 2x+ 4x3 =9 ( 4x3 ) 2 = (92x ) 2 4x3 =8136x+4 x 2 0 =8136x4x+3+4 x 2 4 x 2 40x+84 =0 Dividiendo la ecuación para 4 x 2 10x+21 =0 (x7 ) (x3 ) =0 x7 =0 x 1 =7 x3 =0 x 2 =3 Verificación : Sea x=3 2( 3 ) + 4( 3 ) 3 =3 6+ 123 =3 6+ 9 =3 6+3 =3 9 =3 3 =3x=3  si satisface la ecuación dada Sea x=7 2( 7 ) + 4( 7 ) 3 =3 14+ 283 =3 14+ 25 =3 14+5 =3 19 3x=7  no satisface la ecuación, por tanto es una solución extraña
    10. x+3 + 6 x+3 =5 ( x+3 ) 2 +6 x+3 =5 x+3+6 x+3 =5 (x+9 ) 2 = (5 x+3 ) 2 x 2 +18x+81 =25(x+3 ) x 2 +18x+81 =25x+75 x 2 +18x+8125x75 =0 x 2 7x+6 =0 (x6 ) (x1 ) =0 x6 =0 x 1 =6 x1 =0 x 2 =1 Verificación : Sea x=6 6+3 + 6 6+3 =5 9 + 6 9 =5 3+ 3 =5 3+2 =5 5 =5x=6  si satisface la ecuación dada Sea x=1 1+3 + 6 1+3 =5 4 + 6 4 =5 2+ 2 =5 2+3 =5 5 =5x=1  si satisface la ecuación dada
    11. x + 4 x =5 x+4 x =5 (x+4 ) 2 = (5 x ) 2 x 2 +8x+16 =25x x 2 +8x+1625x =0 x 2 17x+16 =0 (x16 ) (x1 ) =0 x1 =0 x 1 =1 x16 =0 x 2 =16 Verificación : Sea x=1 1 + 4 1 =5 1+4 =5 5 =5x=1  si satisface la ecuación dada Sea x=16 16 + 4 16 =5 4+ 4 4 =5 4+1 =5 5 =5x=1  si satisface la ecuación dada
    12. 2 x = x+7 + 8 x+7 2 x = ( x+7 ) 2 +8 x+7 2 x = x+7+8 x+7 (2 x(x+7 ) ) 2 = (x+15 ) 2 4( x 2 +7x ) = x 2 +30x+225 4 x 2 +28x = x 2 +30x+225 4 x 2 +28x x 2 30x225 =0 3 x 2 2x225 =0 3 x 2 27x+25x225 =0 3x(x9 ) +25(x9 ) =0 (3x+25 ) (x9 ) =0 x9 =0 x 1 =9 3x+25 =0 3x =25 x 2 = 25 3 Verificación : Sea x=9 2 9 = 9+7 + 8 9+7 2( 3 ) = 16 + 8 16 6 =4+ 4 6 =4+2 6 =6x=9  si satisface la ecuación dada Sea x= 25 3 2 25 3 = 25 3 +7 + 8 25 3 +7 2 5 3 i = 25+21 3 + 8 25+21 3 2 5 3 i = 4 3 + 8 4 3 2 5 3 i = 2 3 i + 8 2 3 i 2 5 3 i 2 3 i + 8 3 i 2 x= 25 3  no satisface la ecuación, por tanto es una solución extraña
    13. x+ x+8 =2 x ( x+ x+8 ) 2 = (2 x ) 2 x+ x+8 =4x x+8 =4xx ( x+8 ) 2 = (3x ) 2 x+8 =9 x 2 9 x 2 x8 =0 9 x 2 9x+8x8 =0 9x(x1 ) +8(x1 ) =0 (9x+8 ) (x1 ) =0 x1 =0 x 1 =1 9x+8 =0 9x =8 x 2 = 8 9 Verificación : Sea x=1 1+ 1+8 =2 1 1+ 9 =2 1+3 =2 4 =2 2 =2x=1  si satisface la ecuación dada Sea x= 8 9 8 9 + 8 9 +8 =2 8 9 8 9 + 8+72 9 =2 8 3 i 8 9 + 64 9 =2 8 3 i 8 9 + 8 3 =2 8 3 i 8+24 9 =2 8 3 i 16 9 =2 8 3 i 4 5 2 8 3 ix= 8 9  no satisface la ecuación, por tanto es una solución extraña
    14. 6x + x+7 12x+1 =0 ( 6x + x+7 ) 2 = ( 12x+1 ) 2 6 x +2 (6x ) (x+7 ) + x +7 =12x+1 2 6x+42 x 2 7x =12x+113 2 42 x 2 x =12x12 Dividiendo la ecuación para 2 ( 42 x 2 x ) 2 = (6x6 ) 2 42 x 2 x =36 x 2 72x+36 42 x 2 x36 x 2 +72x36 =0 37 x 2 +71x+6 =0 37 x 2 71x6 =0 37 x 2 74x+3x6 =0 37x(x2 ) +3(x2 ) =0 (37x+3 ) (x2 ) =0 x2 =0 x 1 =2 37x+3 =0 37x =3 x 2 = 3 37 Verificación : Sea x=2 62 + 2+7 12( 2 ) +1 =0 4 + 9 24+1 =0 2+3 25 =0 55 =0 0 =0x=2  si satisface la ecuación dada Sea x= 3 37 6( 3 37 ) + 3 37 +7 12( 3 37 ) +1 =0 6+ 3 37 + 3+259 37 36 37 +1 =0 222+3 37 + 256 37 36+37 37 =0 225 37 + 16 37 37 1 37 =0 15 37 37 + 16 37 37 37 37 =0 37 37 (15+161 ) =0 30 37 37 0x= 3 37  no satisface la ecuación, por tanto es una solución extraña

    Ejercicio 272

    CAPITULO XXXIII

    Ecuaciones incompletas de segundo grado
    Ejercicio 272
    Resolver las ecuaciones:
    1. x 2 =5x x 2 5x =0 x(x5 ) =0 x 1 =0 x5 =0 x 2 =5
    2. 4 x 2 =32x 4 x 2 +32x =0 4x(x+8 ) =0 4x =0 x 1 =0 x+8 =0 x 2 =8
    3. x 2 3x =3 x 2 4x x 2 3 x 2 +4x3x =0 2 x 2 +x =0 x(2x+1 ) =0 x 1 =0 2x+1 =0 2x =1 x 2 = 1 2
    4. 5 x 2 +4 =2(x+2 ) 5 x 2 + 4 =2x+ 4 5 x 2 2x =0 x(5x2 ) =0 x 1 =0 5x2 =0 5x =2 x 2 = 2 5
    5. (x3 ) 2 (2x+5 ) 2 =16 x 2 6x+9(4 x 2 +20x+25 ) =16 x 2 6x+94 x 2 20x25 =16 3 x 2 26x 16 = 16 3 x 2 +26x =0 x(3x+26 ) =0 x 1 =0 3x+26 =0 3x =26 x 2 = 26 3
    6. x 2 3 x9 6 = 3 2 Multiplicando la ecuación por 6 2 x 2 x+ 9 = 9 x(2x1 ) =0 x 1 =0 2x1 =0 2x =1 x 2 = 1 2
    7. (4x1 ) (2x+3 ) =(x+3 ) (x1 ) 8 x 2 +12x2x 3 = x 2 +2x 3 8 x 2 +10x x 2 2x =0 7 x 2 +8x =0 x(7x+8 ) =0 x 1 =0 7x+8 =0 7x =8 x 2 = 8 7
    8. x+1 x1 x+4 x2 =1 (x+1 ) (x2 ) (x+4 ) (x1 ) (x1 ) (x2 ) =1 x 2 x2( x 2 +3x4 ) =(x1 ) (x2 ) x 2 x2 x 2 3x +4 = x 2 3x +2 0 = x 2 +x+ 2 2 0 =x(x+1 ) x 1 =0 0 =x+1 x 2 =1

    Ejercicio 271

    CAPITULO XXXIII

    Ecuaciones incompletas de segundo grado
    Ejercicio 271
    Resolver las ecuaciones:
    1. 3 x 2 =48 x 2 = 3 x = 16 x = ± 4
    2. 5 x 2 9 =46 5 x 2 =46+9 5 x 2 =55 x 2 = 5 x = ± 11
    3. 7 x 2 +14 =0 7( x 2 +2 ) =0 x 2 +2 =0 x 2 =2 x = 2 x = ± 2 i
    4. 9 x 2 a 2 =0 (3x+a ) (3xa ) =0 3x+a =0 3x =a x 1 = a 3 3xa =0 3x =a x 2 = a 3
    5. (x+5 ) (x5 ) =7 x 2 25 =7 x 2 =257 x 2 =18 x = 18 x = 3 2 .2 x = ± 3 2
    6. (2x3 ) (2x+3 ) 135 =0 4 x 2 9135 =0 4 x 2 =144 x 2 = 4 x = 36 x = ± 6
    7. 3(x+2 ) (x2 ) = (x4 ) 2 +8x 3( x 2 4 ) = x 2 8x +16+ 8x 3 x 2 12 = x 2 +16 3 x 2 x 2 =16+12 2 x 2 =28 x 2 = 2 x = ± 14
    8. (x+ 1 3 ) (x 1 3 ) = 1 3 x 2 1 9 = 1 3 x 2 = 1 3 + 1 9 x 2 = 3+1 9 x 2 = 4 9 x = 4 9 x = ± 2 3
    9. (2x1 ) (x+2 ) (x+4 ) (x1 ) +5 =0 2 x 2 +4xx2( x 2 +3x4 ) +5 =0 2 x 2 + 3x 2 x 2 3x +4+5 =0 x 2 +7 =0 x 2 =7 x = 7 x = ± 7 i
    10. 5 2 x 2 1 6 x 2 = 7 12 151 6 x 2 = 7 2 = 7 x 2 x = 4 x = ± 2
    11. 2x3 x3 = x2 x1 (2x3 ) (x1 ) =(x2 ) (x3 ) 2 x 2 2x 3x +3 = x 2 5x +6 2 x 2 x 2 =63 x 2 =3 x = ± 3
    12. x 2 5 3 + 4 x 2 1 5 14 x 2 1 15 =0 Multiplicando la ecuación por 15 5( x 2 5 ) +3(4 x 2 1 ) 14 x 2 +1 =0 5 x 2 25+12 x 2 314 x 2 +1 =0 3 x 2 27 =0 3( x 2 9 ) =0 x 2 9 =0 x 2 =9 x = 9 x = ± 3
    13. 2x3 x 2 +1 x2 =7 (2x3 ) (x2 ) x 2 1 x2 =7 2 x 2 4x3x+6 x 2 1 =7(x2 ) x 2 7x +5 = 7x +14 x 2 =5+14 x 2 =9 x = 9 x = ± 3
    14. 3 3 4 x 2 1 =2 3 4 x 2 1 =23 3 4 x 2 1 =1 3 =4 x 2 1 0 =4 x 2 13 4 x 2 4 =0 4( x 2 1 ) =0 x 2 1 =0 (x1 ) (x+1 ) =0 x1 =0 x 1 =1 x+1 =0 x 2 =1

    Ejercicio 270

    CAPITULO XXXIII

    Ecuaciones literales de segundo grado
    Ejercicio 270
    Resolver las ecuaciones:
    1. x 2 +2ax35 a 2 =0 (x+7a ) (x5a ) =0 x+7a =0 x 1 =7a x5a =0 x 2 =5a
    2. 10 x 2 =36 a 2 37ax 10 x 2 +37ax36 a 2 =0 10 x 2 +45ax8ax36 a 2 =0 5x(2x+9a ) 4a(2x+9a ) =0 (2x+9a ) (5x4a ) =0 2x+9a =0 2x =9a x 1 = 9a 2 5x4a =0 5x =4a x 2 = 4a 5
    3. a 2 x 2 +abx2 b 2 =0 a 2 x 2 abx+2abx2 b 2 =0 ax(axb ) +2b(axb ) =0 (axb ) (ax+2b ) =0 axb =0 ax =b x 1 = b a ax+2b =0 ax =2b x 2 = 2b a
    4. 89bx =42 x 2 +22 b 2 42 x 2 89bx+22 b 2 =0 42 x 2 12bx77bx+22 b 2 =0 6x(7x2b ) 11b(7x2b ) =0 (7x2b ) (6x11b ) =0 7x2b =0 7x =2b x 1 = 2b 7 6x11b =0 6x =11b x 2 = 11b 6
    5. x 2 +ax =20 a 2 x 2 +ax20 a 2 =0 (x+5a ) (x4a ) =0 x+5a =0 x 1 =5a x4a =0 x 2 =4a
    6. 2 x 2 =abx+3 a 2 b 2 2 x 2 abx3 a 2 b 2 =0 2 x 2 +2abx3abx3 a 2 b 2 =0 2x(x+ab ) 3ab(x+ab ) =0 (x+ab ) (2x3ab ) =0 x+ab =0 x 1 =ab 2x3ab =0 2x =3ab x 2 = 3ab 2
    7. b 2 x 2 +2abx =3 a 2 b 2 x 2 +2abx3 a 2 =0 b 2 x 2 abx+3abx3 a 2 =0 bx(bxa ) +3a(bxa ) =0 (bxa ) (bx+3a ) =0 bxa =0 bx =a x 1 = a b bx+3a =0 bx =3a x 2 = 3a b
    8. x 2 +axbx =ab x 2 +axbxab =0 x(x+a ) b(x+a ) =0 (x+a ) (xb ) =0 x+a =0 x 1 =a xb =0 x 2 =b
    9. x 2 2ax =6ab3bx x 2 2ax+3bx6ab =0 x(x2a ) +3b(x2a ) =0 (x2a ) (x+3b ) =0 x2a =0 x 1 =2a x+3b =0 x 2 =3b
    10. 3(2 x 2 mx ) +4nx2mn =0 3x(2xm ) +2n(2xm ) =0 (3x+2n ) (2xm ) =0 3x+2n =0 3x =2n x 1 = 2n 3 2xm =0 2x =m x 2 = m 2
    11. x 2 a 2 bxab =0 (xa ) (x+a ) b(x+a ) =0 (x+a ) (xab ) =0 x+a =0 x 1 =a xab =0 x 2 =a+b
    12. ab x 2 x(b2a ) =2 ab x 2 bx+2ax2 =0 bx(ax1 ) +2(ax1 ) =0 (ax1 ) (bx+2 ) =0 ax1 =0 ax =1 x 1 = 1 a bx+2 =0 bx =2 x 2 = 2 b
    13. x 2 2ax+ a 2 b 2 =0 (xa ) 2 b 2 =0 [(xa ) +b ] [(xa ) b ] =0 (xa+b ) (xab ) =0 xa+b =0 x 1 =ab xab =0 x 2 =a+b
    14. 4x(xb ) + b 2 =4 m 2 4 x 2 4xb+ b 2 4 m 2 =0 (2xb ) 2 4 m 2 =0 [(2xb ) +2m ] [(2xb ) 2m ] =0 (2xb+2m ) (2xb2m ) =0 2xb2m =0 2x =b+2m x 1 = b+2m 2 2xb+2m =0 2x =b2m x 2 = b2m 2
    15. x 2 b 2 +4 a 2 4ax =0 x 2 4ax+4 a 2 b 2 =0 (x2a ) 2 b 2 =0 [(x2a ) +b ] [(x2a ) b ] =0 (x2a+b ) (x2ab ) =0 x2ab =0 x 1 =2a+b x2a+b =0 x 2 =2ab
    16. x 2 (a+2 ) x =2a x 2 ax2x+2a =0 x(xa ) 2(xa ) =0 (xa ) (x2 ) =0 xa =0 x 1 =a x2 =0 x 1 =2
    17. x 2 +2x(43a ) =48a x 2 +8x6ax48a =0 x(x+8 ) 6a(x+8 ) =0 (x+8 ) (x6a ) =0 x+8 =0 x 1 =8 x6a =0 x 2 =6a
    18. x 2 2x = m 2 +2m x 2 m 2 2m2x =0 (x+m ) (xm ) 2(m+x ) =0 (x+m ) [(xm ) 2 ] =0 (x+m ) (xm2 ) =0 x+m =0 x 1 =m xm2 =0 x 2 =m+2
    19. x 2 + m 2 x(m2 ) =2 m 5 x 2 + m 3 x2 m 2 x2 m 5 =0 x(x+ m 3 ) 2 m 2 (x+ m 3 ) =0 (x2 m 2 ) (x+ m 3 ) =0 x+ m 3 =0 x 1 = m 3 x2 m 2 =0 x 2 =2 m 2
    20. 6 x 2 15ax =2bx5ab 6 x 2 2bx15ax+5ab =0 2x(3xb ) 5a(3xb ) =0 (2x5a ) (3xb ) =0 3xb =0 3x =b x 1 = b 3 2x5a =0 2x =5a x 2 = 5a 2
    21. 3x 4 + a 2 x 2 2a =0 Multiplicando la ecuación por 4a 2 x 2 3ax2 a 2 =0 2 x 2 +ax4ax2 a 2 =0 x(2x+a ) 2a(2x+a ) =0 (x2a ) (2x+a ) =0 2x+a =0 2x =a x 1 = a 2 x2a =0 x 2 =2a
    22. 2xb 2 = 2bx b 2 3x 3x(2xb ) =2(2bx b 2 ) 6 x 2 3bx =4bx2 b 2 6 x 2 3bx4bx+2 b 2 =0 6 x 2 7bx+2 b 2 =0 6 x 2 3bx4bx+2 b 2 =0 3x(2xb ) 2b(2xb ) =0 (3x2b ) (2xb ) =0 2xb =0 2x =b x 1 = b 2 3x2b =0 3x =2b x 2 = 2b 3
    23. a+x ax + a2x a+x =4 (a+x ) 2 +(ax ) (a2x ) (ax ) (a+x ) =4 a 2 + 2ax + x 2 + a 2 2ax ax+2 x 2 a 2 x 2 =4 3 x 2 ax+2 a 2 =4( a 2 x 2 ) 3 x 2 ax+2 a 2 =4 a 2 +4 x 2 3 x 2 ax+2 a 2 +4 a 2 4 x 2 =0 x 2 ax+6 a 2 =0 x 2 +ax6 a 2 =0 x 2 2ax+3ax6 a 2 =0 x(x2a ) +3a(x2a ) =0 (x2a ) (x+3a ) =0 x2a =0 x 1 =2a x+3a =0 x 2 =3a
    24. x 2 x1 = a 2 2(a2 ) 2 x 2 (a2 ) = a 2 (x1 ) 2a x 2 4 x 2 = a 2 x a 2 2a x 2 4 x 2 a 2 x+ a 2 =0 2a x 2 a 2 x+ a 2 4 x 2 =0 ax(2xa ) +(a+2x ) (a2x ) =0 ax(a2x ) +(a+2x ) (a2x ) =0 (a2x ) (ax+a+2x ) =0 a2x =0 a =2x x 1 = a 2 ax+a+2x =0 x(2a ) =a x 2 = a 2a a a2
    25. x+ 2 x = 1 a +2a x 2 +2 x = 1+2 a 2 a a( x 2 +2 ) =x(1+2 a 2 ) a x 2 +2a =x+2 a 2 x a x 2 +2ax2 a 2 x =0 a x 2 x2 a 2 x+2a =0 x(ax1 ) 2a(ax1 ) =0 (x2a ) (ax1 ) =0 x2a =0 x 1 =2a ax1 =0 ax =1 x 2 = 1 a
    26. 2xb b x x+b = 2x 4b (2xb ) (x+b ) bx b (x+b ) = 2 x b 2 x 2 + 2bx bx b 2 bx x+b = x 2 2(2 x 2 b 2 ) =x(x+b ) 4 x 2 2 b 2 = x 2 +bx 4 x 2 2 b 2 x 2 bx =0 3 x 2 bx2 b 2 =0 3 x 2 3bx+2bx2 b 2 =0 3x(xb ) +2b(xb ) =0 (3x+2b ) (xb ) =0 xb =0 x 1 =b 3x+2b =0 3x =2b x 2 = 2b 3

    Ejercicio 269

    CAPITULO XXXIII

    Ecuaciones de segundo grado por descomposición en factores
    Ejercicio 269
    Resolver por descomposición en factores:
    1. x 2 x6 =0 (x3 ) (x+2 ) =0 x3 =0 x 1 =3 x+2 =0 x 2 =2
    2. x 2 +7x =18 x 2 +7x18 =0 (x+9 ) (x2 ) =0 x+9 =0 x 1 =9 x2 =0 x 2 =2
    3. 8x65 = x 2 x 2 +8x65 =0 (x+13 ) (x5 ) =0 x+13 =0 x 1 =13 x5 =0 x 2 =5
    4. x 2 =1083x x 2 +3x108 =0 (x+12 ) (x9 ) =0 x+12 =0 x 1 =12 x9 =0 x 2 =9
    5. 2 x 2 +7x4 =0 2 x 2 x+8x4 =0 x(2x1 ) +4(2x1 ) =0 (2x1 ) (x+4 ) =0 x+4 =0 x 1 =4 2x1 =0 2x =1 x 2 = 1 2
    6. 6 x 2 =1011x 6 x 2 +11x10 =0 6 x 2 4x+15x10 =0 2x(3x2 ) +5(3x2 ) =0 (3x2 ) (2x+5 ) =0 3x2 =0 x 1 = 2 3 2x+5 =0 x 2 = 5 2
    7. 20 x 2 27x =14 20 x 2 27x14 =0 20 x 2 35x+8x14 =0 5x(4x7 ) +2(4x7 ) =0 (4x7 ) (5x+2 ) =0 4x7 =0 x 1 = 7 4 5x+2 =0 x 2 = 2 5
    8. 7x =1530 x 2 30 x 2 +7x15 =0 30 x 2 18x+25x15 =0 6x(5x3 ) +5(5x3 ) =0 (5x3 ) (6x+5 ) =0 6x+5 =0 x 1 = 5 6 5x3 =0 x 2 = 3 5
    9. 60 =8 x 2 +157x 8 x 2 +157x60 =0 8 x 2 +160x3x60 =0 8x(x+20 ) 3(x+20 ) =0 (x+20 ) (8x3 ) =0 x+20 =0 x 1 =20 8x3 =0 x 2 = 3 8
    10. x(x1 ) 5(x2 ) =2 x 2 x5x+102 =0 x 2 6x+8 =0 (x4 ) (x2 ) =0 x4 =0 x 1 =4 x2 =0 x 2 =2
    11. (x2 ) 2 (2x+3 ) 2 =80 [(x2 ) +(2x+3 ) ] [(x2 ) (2x+3 ) ] =80 [x2+2x+3 ] [x22x3 ] =80 (3x+1 ) (x5 ) =80 3 x 2 15xx5+80 =0 3 x 2 16x+75 =0 3 x 2 +16x75 =0 3 x 2 9x+25x75 =0 3x(x3 ) +25(x3 ) =0 (x3 ) (3x+25 ) =0 x3 =0 x 1 =3 3x+25 =0 x 2 = 25 3
    12. 6 x 2 9 x = 4 3 69x x 2 = 4 3 3(69x ) =4 x 2 4 x 2 27x+18 =0 4 x 2 24x3x+18 =0 4x(x6 ) 3(x6 ) =0 (x6 ) (4x3 ) =0 x6 =0 x 1 =6 4x3 =0 x 2 = 3 4
    13. x+2 x +x = 74 x x+2+ x 2 x = 74 x x+2+ x 2 74 =0 x 2 +x72 =0 (x+9 ) (x8 ) =0 x8 =0 x 1 =8 x+9 =0 x 2 =9
    14. (x+2 ) 2 2x5 3 =3 3 (x+2 ) 2 2x+5 3 =3 3( x 2 +4x+4 ) 2x+5 =9 3 x 2 +12x+122x+59 =0 3 x 2 +10x+8 =0 3 x 2 +6x+4x+8 =0 3x(x+2 ) +4(x+2 ) =0 (x+2 ) (3x+4 ) =0 x+2 =0 x 1 =2 3x+4 =0 x 2 = 4 3
    15. x x2 +x = 3x+15 4 x+x(x2 ) x2 = 3x+15 4 x+ x 2 2x x2 = 3x+15 4 x 2 x x2 = 3x+15 4 4( x 2 x ) =(x2 ) (3x+15 ) 4 x 2 4x =3 x 2 +15x6x30 4 x 2 4x3 x 2 9x+30 =0 x 2 13x+30 =0 (x10 ) (x3 ) =0 x10 =0 x 1 =10 x3 =0 x 2 =3
    16. 6 x4 4 x = 5 12 6x4(x4 ) x(x4 ) = 5 12 6x4x+16 x 2 4x = 5 12 12(2x+16 ) =5( x 2 4x ) 24x+192 =5 x 2 20x 0 =5 x 2 20x24x192 5 x 2 44x192 =0 5 x 2 60x+16x192 =0 5x(x12 ) +16(x12 ) =0 (x12 ) (5x+16 ) =0 x12 =0 x 1 =12 5x+16 =0 x 2 = 16 5
    17. (x2 ) 3 (x3 ) 3 =37 [(x2 ) (x3 ) ] [ (x2 ) 2 +(x2 ) (x3 ) + (x3 ) 2 ] =37 [ x 2 4x+4+ x 2 5x+6+ x 2 6x+9 ] =37 3 x 2 15x+1937 =0 3 x 2 15x18 =0 Dividiendo la ecuación para 3 x 2 5x6 =0 (x6 ) (x+1 ) =0 x6 =0 x 1 =6 x+1 =0 x 2 =1
    18. x1 x+1 2 = x+3 3 x12(x+1 ) x+1 = x+3 3 x12x2 x+1 = x+3 3 x3 x+1 = x+3 3 3(x3 ) =(x+3 ) (x+1 ) 3x9 = x 2 +4x+3 3x9 x 2 4x3 =0 x 2 7x12 =0 x 2 +7x+12 =0 (x+4 ) (x+3 ) =0 x+4 =0 x 1 =4 x+3 =0 x 2 =3
    19. 4x1 2x+3 = 2x+1 6x+5 (4x1 ) (6x+5 ) =(2x+1 ) (2x+3 ) 24 x 2 +20x6x5 =4 x 2 +8x+3 24 x 2 +14x54 x 2 8x3 =0 20 x 2 +6x8 =0 Dividiendo la ecuación para 2 10 x 2 +3x4 =0 10 x 2 5x+8x4 =0 5x(2x1 ) +4(2x1 ) =0 (2x1 ) (5x+4 ) =0 2x1 =0 x 1 = 1 2 5x+4 =0 x 2 = 4 5
    20. 3x+2 4 =5 9x+14 12x 3x+2 4 = 60x9x14 x 3x+2 = 51x14 3x 3x(3x+2 ) =51x14 9 x 2 +6x51x+14 =0 9 x 2 45x+14 =0 9 x 2 3x42x+14 =0 3x(3x1 ) 14(3x1 ) =0 (3x1 ) (3x14 ) =0 3x1 =0 x 1 = 1 3 3x14 =0 x 2 = 14 3

    Ejercicio 268

    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

    Ejercicio 267

    CAPITULO XXXIII

    Ecuaciones de segundo grado
    Ejercicio 267
    Resolver las siguientes ecuaciones aplicando la fórmula particular:
    x= m 2 ± m 2 4 n
    1. x 2 3x+2 =0 m=3 n=2 x= 3 2 ± (3 ) 2 4 2 x= 3 2 ± 9 4 2 x= 3 2 ± 98 4 x= 3 2 ± 1 4 x= 3 2 ± 1 2 x 1 = 3 2 + 1 2 = 2 x 1 =2 x 2 = 3 2 1 2 = 2 2 x 2 =1
    2. x 2 2x15 =0 m=2 n=15 x= 2 2 ± (2 ) 2 4 (15 ) x=1 ± 4 4 +15 x=1 ± 1+15 x=1 ± 16 x=1 ± 4 x 1 =1+4 x 1 =5 x 2 =14 x 2 =3
    3. x 2 =19x88 x 2 19x+88 =0 m=19 n=88 x= 19 2 ± (19 ) 2 4 88 x= 19 2 ± 361 4 88 x= 19 2 ± 361352 4 x= 19 2 ± 9 4 x= 19 2 ± 3 2 x 1 = 19 2 + 3 2 = 2 x 1 =11 x 2 = 19 2 3 2 = 2 x 2 =8
    4. x 2 +4x =285 x 2 +4x285 =0 m=4 n=285 x= 2 ± 4 2 4 (285 ) x=2 ± 4+285 x=2 ± 289 x=2 ± 17 x 1 =2+17 x 1 =15 x 2 =217 x 2 =19
    5. 5x(x1 ) 2(2 x 2 7x ) =8 5 x 2 5x4 x 2 +14x+8 =0 x 2 +9x+8 =0 m=9 n=8 x= 9 2 ± 9 2 4 8 x= 9 2 ± 81 4 8 x= 9 2 ± 8132 4 x= 9 2 ± 49 4 x= 9 2 ± 7 2 x 1 = 9 2 + 7 2 = 9+7 2 = 2 2 x 1 =1 x 2 = 9 2 7 2 = 97 2 = 2 x 2 =8
    6. x 2 (7x+6 ) =x+59 x 2 7x6x59 =0 x 2 8x65 =0 m=8 n=65 x= 2 ± 8 2 4 (65 ) x=4 ± 4 +65 x=4 ± 16+65 x=4 ± 81 x=4 ± 9 x 1 =4+9 x 1 =13 x 2 =49 x 2 =5
    7. (x1 ) 2 +11x+199 =3 x 2 (x2 ) 2 x 2 2x+1+11x+199 =3 x 2 ( x 2 4x+4 ) x 2 +9x+200 =3 x 2 x 2 +4x4 x 2 +9x+2002 x 2 4x+4 =0 x 2 +5x+204 =0 x 2 5x204 =0 m=5 n=204 x= 5 2 ± (5 ) 2 4 (204 ) x= 5 2 ± 25 4 +204 x= 5 2 ± 25+816 4 x= 5 2 ± 841 4 x= 5 2 ± 29 2 x 1 = 5 2 + 29 2 = 5+29 2 = 2 x 1 =17 x 2 = 5 2 29 2 = 529 2 = 2 x 2 =12
    8. (x2 ) (x+2 ) 7(x1 ) =21 x 2 47x+7 =21 x 2 7x+321 =0 x 2 7x18 =0 m=7 n=17 x= 7 2 ± (7 ) 2 4 (18 ) x= 7 2 ± 49 4 +18 x= 7 2 ± 49+72 4 x= 7 2 ± 121 4 x= 7 2 ± 11 2 x 1 = 7 2 + 11 2 = 7+11 2 = 2 x 1 =9 x 2 = 7 2 11 2 = 711 2 = 2 x 2 =2
    9. 2 x 2 (x2 ) (x+5 ) =7(x+3 ) 2 x 2 ( x 2 +3x10 ) =7x+21 2 x 2 x 2 3x+107x21 =0 x 2 10x11 =0 m=10 n=11 x= 2 ± (10 ) 2 4 (11 ) x=5 ± 4 +11 x=5 ± 25+11 x=5 ± 36 x=5 ± 6 x 1 =5+6 x 1 =11 x 2 =56 x 2 =1
    10. (x1 ) (x+2 ) (2x3 ) (x+4 ) x+14 =0 x 2 + x 2(2 x 2 +8x3x12 ) x +14 =0 x 2 2 x 2 5x+12+12 =0 x 2 5x+24 =0 x 2 +5x24 =0 m=5 n=24 x= 5 2 ± 5 2 4 (24 ) x= 5 2 ± 25 4 +24 x= 5 2 ± 25+96 4 x= 5 2 ± 121 4 x= 5 2 ± 11 2 x 1 = 5 2 + 11 2 = 5+11 2 = 2 x 1 =3 x 2 = 5 2 11 2 = 511 2 = 2 x 2 =8

    Ejercicio 266

    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

    Ejercicio 265

    CAPITULO XXXIII

    Ecuaciones de segundo grado
    Ejercicio 265
    Resolver las siguientes ecuaciones por la fórmula general:
    x= b ± b 2 4ac 2a
    1. 3 x 2 5x+2 =0 x = 5 ± 5 2 4( 3 ) ( 2 ) 2( 3 ) x = 5 ± 2524 6 x = 5 ± 1 6 x 1 = 6 6 x 1 =1 x 2 = x 2 = 2 3
    2. 4 x 2 +3x22 =0 x = 3 ± 3 2 4( 4 ) (22 ) 2( 4 ) x = 3 ± 9352 8 x = 3 ± 361 8 x = 3 ± 19 8 x 1 = 8 x 1 =2 x 2 = x 2 = 11 4
    3. x 2 +11x =24 x 2 +11x+24 =0 x = 11 ± 1 1 2 4( 1 ) ( 24 ) 2( 1 ) x = 11 ± 12196 2 x = 11 ± 25 2 x = 11 ± 5 2 x 1 = 2 x 1 =8 x 2 = 2 x 2 =3
    4. x 2 =16x63 x 2 16x+63 =0 x = 16 ± 1 6 2 4( 1 ) ( 63 ) 2( 1 ) x = 16 ± 1 6 2 4( 1 ) ( 63 ) 2( 1 ) x = 16 ± 256252 2 x = 16 ± 4 2 x = 16 ± 2 2 x 1 = 2 x 1 =9 x 2 = 2 x 2 =7
    5. 12x49 x 2 =0 9 x 2 12x+4 =0 x = 12 ± 1 2 2 4( 9 ) ( 4 ) 2( 9 ) x = 12 ± 18 x = x = 2 3
    6. 5 x 2 7x90 =0 x = 7 ± 7 2 4( 5 ) (90 ) 2( 5 ) x = 7 ± 49+1800 10 x = 7 ± 1849 10 x = 7 ± 43 10 x 1 = 5 0 1 0 x 1 =5 x 2 = x 2 = 18 5
    7. 6 x 2 =x+222 6 x 2 x222 =0 x = 1 ± 14( 6 ) (222 ) 2( 6 ) x = 1 ± 1+5328 12 x = 1 ± 73 12 x 1 = x 1 = 37 6 x 2 = 12 x 2 =6
    8. x+11 =10 x 2 10 x 2 x11 =0 x = 1 ± 14( 10 ) (11 ) 2( 10 ) x = 1 ± 1+440 20 x = 1 ± 21 20 x 1 = x 1 = 11 10 x 2 = 20 20 x 2 =1
    9. 49 x 2 70x+25 =0 x = 70 ± 7 0 2 4( 49 ) ( 25 ) 2( 49 ) x = 70 ± 98 x = x = 5 7
    10. 12x7 x 2 +64 =0 7 x 2 12x64 =0 x = 12 ± 1 2 2 4( 7 ) (64 ) 2( 7 ) x = 12 ± 144+1792 14 x = 12 ± 1936 14 x = 12 ± 44 14 x 1 = 14 x 1 =4 x 2 = x 2 = 16 7
    11. x 2 =15x56 x 2 +15x+56 =0 x = 15 ± 1 5 2 4( 1 ) ( 56 ) 2( 1 ) x = 15 ± 225224 2 x = 15 ± 1 2 x 1 = 2 x 1 =7 x 2 = 2 x 2 =8
    12. 32 x 2 +18x17 =0 x = 18 ± 1 8 2 4( 32 ) (17 ) 2( 32 ) x = 18 ± 324+2176 64 x = 18 ± 2500 64 x = 18 ± 50 64 x 1 = 32 x 1 = 1 2 x 2 = x 2 = 17 16
    13. 176x =121+64 x 2 64 x 2 176x+121 =0 x = 176 ± 17 6 2 4( 64 ) ( 121 ) 2( 64 ) x = 176 ± 128 x = x = 11 8
    14. 8x+5 =36 x 2 36 x 2 8x5 =0 x = 8 ± 8 2 4( 36 ) (5 ) 2( 36 ) x = 8 ± 64+720 72 x = 8 ± 784 72 x = 8 ± 28 72 x 1 = 36 x 1 = 1 2 x 2 = x 2 = 5 18
    15. 27 x 2 +12x7 =0 x = 12 ± 1 2 2 4( 27 ) (7 ) 2( 27 ) x = 12 ± 144+756 54 x = 12 ± 900 54 x = 12 ± 30 54 x 1 = 18 x 1 = 1 3 x 2 = x 2 = 7 9
    16. 15x =25 x 2 +2 25 x 2 15x+2 =0 x = 15 ± 1 5 2 4( 25 ) ( 2 ) 2( 25 ) x = 15 ± 225200 50 x = 15 ± 25 50 x = 15 ± 5 50 x 1 = 2 0 5 0 x 1 = 2 5 x 2 = 1 0 5 0 x 2 = 1 5
    17. 8 x 2 2x3 =0 x = 2 ± 2 2 4( 8 ) (3 ) 2( 8 ) x = 2 ± 4+96 16 x = 2 ± 100 16 x = 2 ± 10 16 x 1 = x 1 = 3 4 x 2 = 8 x 2 = 1 2
    18. 105 =x+2 x 2 2 x 2 +x105 =0 x = 1 ± 14( 2 ) (105 ) 2( 2 ) x = 1 ± 14( 2 ) (105 ) 4 x = 1 ± 1+840 4 x = 1 ± 841 4 x = 1 ± 29 4 x 1 = 4 x 1 =7 x 2 = x 2 = 15 2