solve-this-equation-in-all-part-of-complex-number-x-9-3x-2-1-x-6-4-x-9-3x-2-1-x-6-16-

Question Number 63383 by minh2001 last updated on 03/Jul/19

s o l v e t h i s e q u a t i o n i n a l l

p a r t o f c o m p l e x n u m b e r :

\sqrt{(x^{9} - 3 x^{2} + 1) (x - 6) + 4} = (x^{9} - 3 x^{2} + 1) (x - 6) - 16

Commented by MJS last updated on 04/Jul/19

waiting to see your solution

Answered by MJS last updated on 03/Jul/19

\sqrt{t + 4} = t - 16

t + 4 = {(t - 16)}^{2}

t^{2} - 33 t + 252 = 0

\Rightarrow t = 12 \lor t = 21

but t = 12 is no solution of \sqrt{t + 4} = t - 16

\Rightarrow t = 21

(x^{9} - 3 x^{2} + 1) (x - 6) = 21

(x + 1) (x^{9} - 7 x^{8} + 7 x^{7} - 7 x^{6} + 7 x^{5} - 7 x^{4} + 7 x^{3} - 10 x^{2} + 28 x - 27) = 0

x_{1} = - 1

we cannot exactly solve the rest

x_{2} \approx 6.00000208

the other 8 roots are complex (4 pairs of conjugated complex roots)

if {it}^{'} s

\sqrt{(x^{3} - 3 x^{2} + 1) (x - 6) + 4} = (x^{3} - 3 x^{2} + 1) (x - 6) - 16

we can :

(x^{3} - 3 x^{2} + 1) (x - 6) = 21

(x + 1) (x^{3} - 10 x^{2} + 28 x - 27) = 0

x_{1} = - 1

x_{2} = \frac{10}{3} + \sqrt[3]{\frac{209}{54} - \frac{\sqrt{337}}{6}} + \sqrt[3]{\frac{209}{54} + \frac{\sqrt{337}}{6}}

x_{3} = \frac{10}{3} + (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) \sqrt[3]{\frac{209}{54} - \frac{\sqrt{337}}{6}} + (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) \sqrt[3]{\frac{209}{54} + \frac{\sqrt{337}}{6}}

x_{4} = \frac{10}{3} + (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) \sqrt[3]{\frac{209}{54} - \frac{\sqrt{337}}{6}} + (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) \sqrt[3]{\frac{209}{54} + \frac{\sqrt{337}}{6}}

Commented by minh2001 last updated on 04/Jul/19

N o, y o u w e r e w r o n g . W h y

y o u s e t x^{3} b u t n o t x^{9} . I^{'} v e

f o u n d o t h e r 8 s o l u t i o n s a s

y o u r m e a n, t h a n k y o u

Commented by MJS last updated on 04/Jul/19

so show us how you found them

I can approximate them all

Commented by MJS last updated on 04/Jul/19

with x^{9}

x_{1} = - 1

x_{2} \approx 6.00000208383

x_{3, 4} \approx - . 994927692081 \pm . 650751638656 i

x_{5, 6} \approx - . 262576070729 \pm 1 . 25073913893 i

x_{7, 8} \approx . 684103814572 \pm 1 . 04916880346 i

x_{9, 10} \approx 1.07339890633 \pm . 300735402858 i

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