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Could you explain to me please, how to solve the following system of $ 2 $ equations in $ \mathbb{R} $ or $ \mathbb{C} $ : $$ \begin{cases} \ \ \ ab^2 = 2 \\ a+b = -1 \end{cases} $$ According to my opinion, this system seems to correspond to a system in the form : $$ \begin{cases} \ \ \ ab = 2 \\ a+b = -1 \end{cases} $$ which we can solve by solving the following equation of second degree : $ x^2 + x + 2 = 0 $

But here, there is a number in exponent of $ b$ : $ 2 $ in $ ab^2 = 2 $, which doesn't help me to solve the system. Can you help me please ?

Thank you.

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Replace $b$ with $-1-a$ in the first equation (because $a+b=-1$ is given), we will have

\begin{equation}\begin{aligned} a(-1-a)^2=2&\Leftrightarrow a(a^2+2a+1)=2 \\ &\Leftrightarrow a^3+2a^2+a-2=0 \\ \end{aligned}\end{equation}

You need to solve a cubic equation of third degree, not second degree.

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  • $\begingroup$ Thank you very much. :-) $\endgroup$ – YoYo Apr 29 '18 at 11:47
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    $\begingroup$ There is an error in the second line: it should be $+2a^2$, not minus. $\endgroup$ – Ennar Apr 29 '18 at 11:51
  • $\begingroup$ I have edited, however the solution will be quite ugly: $a=0.6956207696; b=-1.6956207696$. $\endgroup$ – user061703 Apr 29 '18 at 11:55
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    $\begingroup$ closed forms: wolframalpha.com/input/?i=solve+a%5E3%2B2a%5E2%2Ba-2%3D0+for+a $\endgroup$ – Jam Apr 29 '18 at 11:56
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    $\begingroup$ @Jam For anyone who can't access to the link: $a=\frac{1}{3}(-2+\sqrt[3]{28-3\sqrt{87}}+\sqrt[3]{28+3\sqrt{87}})$ $\endgroup$ – user061703 Apr 29 '18 at 11:58

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