Quadratische Gleichungen mit quadratischer Ergänzung auflösen: k) -4x^2-16x=-308

Question

g) (x+12)^2-27=457
h) (h-14)h^2+45=301
i) -x^2+9x=14
j) x(9x-6)+1=0
k) -4x^2-16x=-308

Answer 1

(x + 12)^2 - 27 = 457
(x + 12)^2 = 484
(x + 12) = ± 22
x = - 12 ± 22

 

(h - 14)^2 + 45 = 301
(h - 14)^2 = 256
h - 14 = ± 16
h = 14 ± 16

 

-x^2 + 9x = 14
x^2 - 9x = - 14
x^2 - 9x + (9/2)^2 = - 14 + (9/2)^2
(x - 9/2)^2 = 25/4
x - 9/2 = ± 5/2
x = 9/2 ± 5/2

 

x(9x - 6) + 1 = 0
9x^2 - 6x + 1 = 0
x^2 - 2/3*x + 1/9 = 0
x^2 - 2/3*x = - 1/9
x^2 - 2/3*x + (1/3)^2 = - 1/9 + (1/3)^2
(x - 1/3)^2 = 0
x - 1/3 = 0
x = 1/3

 

-4x^2 - 16x = -308
x^2 + 4x = 77
x^2 + 4x + 2^2 = 77 + (4/2)^2
(x + 2)^2 = 81
x + 2 = ± 9
x = - 2 ± 9