Find-all-complex-number-z-such-that-3z-1-4z-1-6z-1-12z-1-2-

Question Number 152757 by EDWIN88 last updated on 01/Sep/21

F i n d a l l c o m p l e x n u m b e r z s u c h

t h a t (3 z + 1) (4 z + 1) (6 z + 1) (12 z + 1) = 2

Answered by john_santu last updated on 01/Sep/21

n o t e t h a t 8 (3 z + 1) 6 (4 z + 1) 4 (6 z + 1) 2 (12 z + 1) = 768

(24 z + 8) (24 z + 6) (24 z + 4) (24 z + 2) = 768

l e t u = 24 z + 5

\Rightarrow (u + 3) (u + 1) (u - 1) (u - 3) = 768

\Rightarrow (u^{2} - 1) (u^{2} - 9) = 768

i . e w^{2} - 10 w - 759 = 0, w = u^{2}

\to {\begin{cases} w_{1} = 33 \Rightarrow z = \frac{\pm \sqrt{33} - 5}{24} \\ w_{2} = - 23 \Rightarrow z = \frac{\pm \sqrt{23 i} - 5}{24} \end{cases}