Question

x^4+4x^3+2x^2-4x+1=0

Answer 1

 

$\displaystyle\bf\\x^4+4x^3+2x^2-4x+1=0\\\\4x^3 = 2x^3 + 2x^3\\2x^2 = -x^2 + 4x^2 - x^2\\-4x = -2x - 2x\\\\x^4+~~~~~2x^3+2x^3~~~~~-x^2 + 4x^2 - x^2~~~~~-2x - 2x~~~~~+1=0\\\\x^4+2x^3-x^2~~~+2x^3+4x^2-2x~~~-x^2-2x+1=0\\\\\text{Din fiecare grupa de 3 termeni dam factor comun.}\\\\x^2\Big(x^2+2x-1\Big)+2x\Big(x^2+2x-1\Big)-1\Big(x^2+2x-1\Big)=0\\\\\Big(x^2+2x-1\Big)\Big(x^2+2x-1\Big)=0$

.

$\displaystyle\bf\\\Big(x^2+2x-1\Big)^2=0\\\\x_{1234}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-2\pm\sqrt{4+4}}{2}=\frac{-2\pm2\sqrt{2}}{2}=\boxed{\bf-1\pm\sqrt{2}}\\\\\\\boxed{\bf~x_1=x_2=-1-\sqrt{2}}\\\boxed{\bf~x_3=x_4=-1+\sqrt{2}}$