x-y-3-1-3-1-x-x-2-y-10-find-solution-

Question Number 82067 by john santu last updated on 18/Feb/20

{\begin{cases} ∣ x ∣ - \sqrt[3]{y + 3} = 1 \\ {(- x \sqrt{- x})}^{2} = y + 10 \end{cases}

f i n d s o l u t i o n

Commented by john santu last updated on 18/Feb/20

Commented by MJS last updated on 18/Feb/20

{there}^{'} s also a complex solution :

x = 1 \pm i \sqrt{3}

y = - 2

Commented by john santu last updated on 18/Feb/20

w h e r e d i d y o u g e t i t ? {i s n}^{'} t t h e c o n d i t i o n

x m u s t b e ⩽ 0 ?

Commented by MJS last updated on 18/Feb/20

x ⩽ 0 if x \in R

if x \in C \Rightarrow \sqrt{- x} always exists

Commented by john santu last updated on 18/Feb/20

y e s s i r . y o u r r i g h t

Answered by MJS last updated on 18/Feb/20

{\begin{cases} y =∣ x ∣^{3} - 3 ∣ x ∣^{2} + 3 ∣ x ∣ - 4 \Rightarrow y \in R \forall x \in C \\ y = - x^{3} - 10 \end{cases}

\Rightarrow - x^{3} - 10 \in R

∣ x ∣^{3} - 3 ∣ x ∣^{2} + 3 ∣ x ∣ - 4 = - x^{3} - 10

let x = a + b i

\sqrt{a^{2} + b^{2}} (a^{2} + b^{2} + 3) + a^{3} - 3 (a^{2} + {a b}^{2} + b^{2} - 2) + i (3 a^{2} - b^{2}) b = 0

{\begin{cases} \sqrt{a^{2} + b^{2}} (a^{2} + b^{2} + 3) + a^{3} - 3 (a^{2} + {a b}^{2} + b^{2} - 2) = 0 \\ (3 a^{2} - b^{2}) b = 0 \Rightarrow b = 0 \lor b = \pm a \sqrt{3} \end{cases}

(1) b = 0

∣ a ∣^{3} + a^{3} - 3 a^{2} + 3 ∣ a ∣ + 6 = 0

(1.1) a > 0

2 a^{3} - 3 a^{2} + 3 a + 6 = 0 \Rightarrow a \approx - . 854 no solution

(1.2) a < 0

- 3 a^{2} - 3 a + 6 = 0 \Rightarrow a = - 2 \lor a = 1 \Rightarrow a = - 2

x_{1} = - 2 \land y_{1} = - 2

(2) b = \pm a \sqrt{3}

8 ∣ a ∣^{3} - 8 a^{3} - 12 a^{2} + 6 ∣ a ∣ + 6 = 0

(2.1) a < 0

- 16 a^{3} - 12 a^{2} - 6 a + 6 = 0 \Rightarrow a \approx . 427 no solution

(2.2) a > 0

- 12 a^{2} + 6 a + 6 = 0 \Rightarrow - \frac{1}{2} \lor a = 1 \Rightarrow a = 1

x_{2, 3} = 1 \pm i \sqrt{3} \land y_{2, 3} = - 2

Commented by john santu last updated on 18/Feb/20

o k a y s i r . i a g r e e y o u r w a y

Commented by MJS last updated on 18/Feb/20

just for fun, I always try to find complex

solutions too

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