Two equations In : C

**dhiab** · July 21st 2009, 02:24 AM

Solve in C :

**DeMath** · July 21st 2009, 03:08 AM

Originally Posted by dhiab

Solve in C :

${z^4} + {\left( {1 + z} \right)^4} = 0 \Leftrightarrow$

$\Leftrightarrow {z^4} + 2{z^2}{\left( {1 + z} \right)^2} + {\left( {1 + z} \right)^4} - 2{z^2}{\left( {1 + z} \right)^2} = 0 \Leftrightarrow$

$\Leftrightarrow {\left( {{z^2} + {{\left( {1 + z} \right)}^2}} \right)^2} - 2{z^2}{\left( {1 + z} \right)^2} = 0 \Leftrightarrow$

$\Leftrightarrow \left( {{z^2} + {{\left( {1 + z} \right)}^2} - \sqrt 2 z\left( {1 + z} \right)} \right)\left( {{z^2} + {{\left( {1 + z} \right)}^2} + \sqrt 2 z\left( {1 + z} \right)} \right) = 0.$

What you need to do now, I hope you know

**Opalg** · July 21st 2009, 03:29 AM

Another method: $z^4 + (1 + z)^4 = 0\: \Leftrightarrow\: (1+z)^4 = (-1)z^4\: \Leftrightarrow\: 1+z = \omega z \: \Leftrightarrow\: z = \frac1{\omega-1}$ , where $\omega$ is one of the fourth roots of –1, namely $\tfrac1{\sqrt2}(\pm1\pm i)$ .

**dhiab** · July 21st 2009, 07:20 AM

Originally Posted by Opalg

Another method: $z^4 + (1 + z)^4 = 0\: \Leftrightarrow\: (1+z)^4 = (-1)z^4\: \Leftrightarrow\: 1+z = \omega z \: \Leftrightarrow\: z = \frac1{\omega-1}$ , where $\omega$ is one of the fourth roots of –1, namely $\tfrac1{\sqrt2}(\pm1\pm i)$ .

Hello thank you we have anathor solution :
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