Question

solve this question which is from complex number ​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{x^9+x^5-x^4=1}$

$\underline{\textbf{To find:}}$

$\textsf{The solution of the given equation}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{x^9+x^5-x^4=1}$

$\mathsf{x^9+x^5-x^4-1=0}$

$\textsf{This can be written as}$

$\mathsf{x^5(x^4+1)-1(x^4+1)=0}$

$\mathsf{(x^5-1)(x^4+1)=0}$

$\implies\mathsf{x^5-1=0\;\;(or)\;\;x^4+1=0}$

$\underline{\mathsf{Case(i):\;}}$

$\mathsf{x^5=1}$

$\mathsf{x^5=cos\,0+i\,sin \,0}$

$\mathsf{x^5=cos\,2k\pi+i\,sin \,2k\pi}$

$\implies\mathsf{x=(cos\,2k\pi+i\,sin\,2k\pi)^{\dfrac{1}{5}}\;\;k=0,1,2,3,4}$

$\textsf{By Demoivre's theorem}$

$\implies\mathsf{x=cos\dfrac{2k\pi}{5}+i\,sin\dfrac{2k\pi}{5}\;\;\;k=0,1,2,3,4}$

$\textsf{The solutions are}$

$\mathsf{For\;k=0,\;\;x=cos\,0+i\,sin\,0}$

$\mathsf{For\;k=1,\;x=cos\dfrac{2\pi}{5}+i\,sin\dfrac{2\pi}{5}}$

$\mathsf{For\;k=2,\;x=cos\dfrac{4\pi}{5}+i\,sin\dfrac{4\pi}{5}}$

$\mathsf{For\;k=3,\;x=cos\dfrac{6\pi}{5}+i\,sin\dfrac{6\pi}{5}}$

$\mathsf{For\;k=4,\;x=cos\dfrac{8\pi}{5}+i\,sin\dfrac{8\pi}{5}}$

$\underline{\mathsf{Case(ii):\;}}$

$\mathsf{x^4+1=0}$

$\mathsf{x^4=-1}$

$\mathsf{x^4=cos\,\pi+i\,sin \,\pi}$

$\mathsf{x^4=cos\,\pi+2k\pi+i\,sin \,\pi+2k\pi}$

$\mathsf{x^4=cos(2k+1)\pi+i\,sin(2k+1)\pi}$

$\implies\mathsf{x=(cos(2k+1)\pi+i\,sin(2k+1)\pi)^{\dfrac{1}{4}}\;\;k=0,1,2,3}$

$\textsf{By Demoivre's theorem}$

$\implies\mathsf{x=cos\dfrac{(2k+1)\pi}{4}+i\,sin\dfrac{(2k+1)\pi}{4}\;\;\;k=0,1,2,3}$

$\textsf{The solutions are}$

$\mathsf{For\;k=0,\;x=cos\dfrac{\pi}{4}+i\,sin\dfrac{\pi}{4}}$

$\mathsf{For\;k=1,\;x=cos\dfrac{3\pi}{4}+i\,sin\dfrac{3\pi}{4}}$

$\mathsf{For\;k=2,\;x=cos\dfrac{5\pi}{4}+i\,sin\dfrac{5\pi}{4}}$

$\mathsf{For\;k=3,\;x=cos\dfrac{7\pi}{4}+i\,sin\dfrac{7\pi}{4}}$

$\therefore\textsf{Totally we have 9 solutions}$

Answer 2

Answer:

The answe is 9

Step-by-step explanation:

i hope it help you.