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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